E HISTORY
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CALCULOS
(THE CONCEPTS OF THE CALCULUS)
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The History of the Calculus
and its
Conceptual Development
(The Concepts of the Calculus)
By
CARL B. BOYER
with a Foreword by
RICHARD COURANT
DOVER PUBLICATIONS, INC., NEW YORK
Copyright © 1949 by Carl B. Boyer.
All rights reserved under Pan American and In-
ternational Copyright Conventions,
Published in Canada by General Publishing Com-
pany, Ltd., 30 Lesmill Road, Don Mills, Toronto,
Ontario,
Published in the United Kingdom by Constable
and Company, Ltd,. 10 Orange Street, London
WC2.
This Dover edition, first published in 1959, is an
unabridged and unaltered republication of the work
originally published by Hafncr Publishing Company,
Inc., in 1949 under the title The Concepts of the.
Calculus, A Critical and Historical Discussion of the
Derivative and the Integral.
Standard Book Number; -f 86-60509-4
Library of Congress Catalog Card Number: 59-9673
Manufactured in the United States of America
Dover Publications, Inc.
180 Variek Street
New York, N, Y. 10014
Foreword
Differential and Integral Calculus and Mathematical
Analysis in general is one of the great achievements of the
human mind. Its place between the natural and humanistic sciences
should make it a singularly fruitful medium of higher education.
Unfortunately, the mechanical way in which calculus sometimes is
taught fails to present the subject as the outcome of a dramatic
intellectual struggle which has lasted for twenty-five hundred years
or more, which is deeply rooted in many phases of human endeavors
and which will continue as long as man strives to understand himself
as well as nature. Teachers, students, and scholars who really want
to comprehend the forces and appearances of science must have some
understanding of the present aspect of knowledge as a result of his-
torical evolution. As a matter of fact, reaction against dogmatism
in scientific teaching has aroused a growing interest in history of
science; during the recent decades very great progress has been made
in tracing the historical roots of science in general and mathematics
in particular.
The present volume, which fortunately can appear in a second
printing, is an important contribution towards clarification of the
many steps which led to the development of the concepts of calculus
from antiquity to the present day; beyond that, it gives a connected
and highly readable account of this fascinating story. The book
ought to reach every teacher of mathematics; then it certainly will
have a strong influence towards a healthy reform in the teaching of
mathematics,
R. COUKANT
Chairman of the Mathematics Department
Graduate School, New York University
Preface
SOME ten years ago Professor Frederick Barry, of Columbia
University, pointed out to me that the history of the calculus
had not been satisfactorily written. Other duties and inadequate
preparation at the time made it impossible to act upon this sugges-
tion, but my studies of the past several years have confirmed this view.
There is indeed no lack of material on the origin and subject matter
of the calculus, as the titles in the bibliography appended to this work
will attest. What is wanting is a satisfactory critical account of the
filiation of the fundamental ideas of the subject from their incipiency
in antiquity to the final formulation of these in the precise concepts
familiar to every student of the elements of modern mathematical
analysis. The present work is an attempt to supply, in some measure,
this deficiency. An authoritative and comprehensive treatment of the
whole history of the elementary calculus is greatly to be desired; but
any such ambitious project is far beyond the scope and intention of the
dissertation here presented. This is not a history of the calculus in
all its aspects, but a suggestive outline of the development of the
basic concepts, and as such should be of service both to students of
mathematics and to scholars in the field of the history of thought.
The aim throughout has therefore been to secure clarity of exposition,
rather than to present a confusingly elaborate all-inclusiveness of
detail or to display a meticulously precise erudition. This has neces-
sitated a judicious selection and presentation of such material as
would preserve the continuity of thought, but it is to be hoped that
historical accuracy and perspective have not thereby been sacrificed.
The inclusion at the end of this volume of an extensive bibliography
of works to which reference has been made has caused it to appear
unnecessary to give full citations in the footnotes. In these notes
author and title— in some cases abbreviated— alone have been given;
titles of books have been italicized, those of articles in the periodical
literature appear in Roman type enclosed within quotation marks. It is
felt that the inclusion of such a list of sources on the subject may serve
to encourage further investigations into the history of the calculus.
Pref
ace
The inspiration toward the projection and completion of the present
study has been due to Professor Barry, who has generously assisted
in its prosecution through advice based on his wide familiarity with
the field of the history of science. Professor Lynn Thorndike, of
Columbia University, very kindly read and offered his competent
criticism of the chapter on "Medieval Contributions." Professors L.
P. SicelofT, of Columbia University, L. C. Karpinski, of the University
of Michigan, and H. F. MacNeish, of Brooklyn College, have also
read the manuscript and have furnished valuable aid and suggestions,
Mrs. Boyer has been unstinting in her encouragement and promotion
of the work, and has painstakingly done all of the typing. The com-
position of the Index has been undertaken by the Columbia Univer-
sity Press. Finally, from the American Council of Learned Societies
came the subvention, in the form of a grant in aid of publication,
which has made possible the appearance of this book in its present
form. To all who have thus contributed toward the preparation and
publication of this volume I wish to express my sincere appreciation.
Cakl B. Boyer
Brooklyn College
January 3, 1939
Preface to the Second Printing
It is gratifying to find sufficient demand for a work on the history
of the calculus to warrant a second printing. This appears to be
a token of the increasing awareness in academic circles of the need
for a broad outlook with respect to science and mathematics. Amaz-
ing achievements in technology notwithstanding, there is a keener
appreciation of the fact that science is a habit of mind as well as a
way of life, and that mathematics is an aspect of culture as well as a
collection of algorithms. The history of these subjects can never be
a substitute for work in the laboratory or for training in techniques,
but it can serve effectively to alleviate the lack of mutual understand-
ing too often existing between the humanities and the sciences. Per-
haps even more important is the role that the history of mathematics
and science can play in the cultivation among professional workers
in the fields of a sense of proportion with respect to their subjects.
No scholar familiar with the historical background of his specialty is
likely to succumb to that specious sense of finality which the novitiate
all too frequently experiences. For this reason, if for no other, it
would be wise for every prospective teacher to know not only the
material of his field but also the story of its development.
In this new printing a few minor errors in the text have been cor-
rected. Were it a new edition, more extensive alterations might have
been justified. These would not have changed substantially the
general account or the point of view; but the argument would have
been clarified along the lines suggested by the judicious reviews of
Julio Rey Pastor (ArcheloiK XXIII [1940] 199-203), L B. Cohen (I sis
XXXII 11940] 205-210), and others. Additional bibliographical
references could have been added, of which one in particular deserves
to be noted here — G. Castelnuovo, Lc origini del calcolo infinites*-
male neW era moderna (Bologna, 1938). Castelnuovo's book, which
appeared at about the same time as the present work, should be
consulted for further details on the modern period written by a
celebrated geometer.
The author has been engaged for the past several years in the
Preface
preparation of a sort of companion volume on the history of analytic
geometry. The manuscript of this work has been completed and
the book should appear before long under the auspices of Scripta
Matkematica.
The reappearance of The Concepts of tite Calcidus> which has been
out of print for well over half a dozen years, is due to Herbert Axel rod
and Martin N. Wright; and the author wishes to express his appre-
ciation of their initiative in making the republication possible. And
to Professor Richard Courant the writer acknowledges a debt of
gratitude in view of the fact that he graciously consented to write a
foreword for the new printing.
Cael B. Boyer
January 27, 1949
Contents
1
I. INTRODUCTION
II, CONCEPTIONS IN ANTIQUITY 14
III. MEDIEVAL CONTRIBUTIONS 61
IV. A CENTURY OF ANTICIPATION 96
V. NEWTON AND LEIBNIZ 187
VI. THE PERIOD OF INDECISION 224
VII. THE RIGOROUS FORMULATION 267
VIII. CONCLUSION 2 "
BIBLIOGRAPHY 311
INDEX 337
The History of the Calculus
I. Introduction
MATHEMATICS has been an integral part of man's intellectual
training and heritage for at least twenty-five hundred years.
During this long period of time, however, no general agreement has
been reached as to the nature of the subject, nor has any universally
acceptable definition been given for it. 1
From the observation of nature, the ancient Babylonians and Egyp-
tians built up a body of mathematical knowledge which they used in
making further observations. Thales perhaps introduced deductive
methods; certainly the mathematics of the early Pythagoreans was
deductive in character. The Pythagoreans and Plato 2 noted that the
conclusions they reached deductively agreed to a remarkable extent
with the results of observation and inductive inference. Unable to
account otherwise for this agreement, they were led to regard mathe-
matics as the study of ultimate, eternal reality, immanent in nature
and the universe, rather than as a branch of logic or a tool of science
and technology. An understanding of mathematical principles, they
decided, must precede any valid interpretation of experience. This
view is reflected in the Pythagorean dictum that all is number, 3 and
in the assertion attributed to Plato that God always plays the
geometer. 4
Later Greek skeptics, it is true, questioned the possibility of
attaining any knowledge of such absolute character either by reason
or by experience. But Aristotelian science had meanwhile shown that
through observation and logic one can at least reach a consistent
representation of phenomena, and mathematics consequently became;
with Euclid, an idealized pattern of deductive relationships. Derived
from postulates consistent with the results of induction from observa-
tion, it was found serviceable in the interpretation of nature.
i Bell, The Queen of the Sciences, p. 15. For full citations of works referred to in the
footnotes see the Bibliography.
* See, for example, Republic VII, 527, in Dialogues, Jowett trans., Vol. II, pp. 362-63.
» See Aristotle, Metaphysics 987a-989b, in Works, ed. by Ross and Smith, Vol. VIII;
Cf. also 1090a.
* Plutarch, Miscellanies and Essays, III, 402.
2 Introduction
The Scholastic view, which prevailed during the Middle Ages, was
that the universe is "tidy" and simply intelligible. In the fourteenth
century came a fairly clear realization that Peripatetic qualitative
views of motion and variation could better be replaced by quantitative
study. These two concepts, with a revival of interest in Platonic views,
brought about in the fifteenth and sixteenth centuries a renewal of the
conviction that mathematics is in some way independent of, and prior
to, experiential and intuitive knowledge. Such conviction is marked in
the thinking of Nicholas of Cusa, Kepler, and Galileo, and to a certain
extent appears in that of Leonardo da Vinci.
This conception of mathematics as the basis of the architecture
of the universe was in turn modified in the sixteenth and seventeenth
centuries. In mathematics, the cause of the change was the less critical
and more practical use of the algebra which had been adopted from the
Arabs, early in the thirteenth century, and then further developed in
Italy. In natural science, the change was due to the rise of experimental
method. The certitude in mathematics of which Descartes, Boyle, and
others spoke was thus interpreted to mean a consistency to be found
rather in the character of its reasoning than in any ontological neces-
sity which it presented a priori.
The centering of attention on the procedures rather than on the
bases of mathematics was emphasized in the eighteenth century by
an extraordinary success in applying the calculus to scientific and
mathematical problems. A more critical attitude was inaugurated in
the nineteenth century by persistent efforts to find a satisfactory
foundation for the conceptions involved in this new analysis of the
infinite. Mathematical rigor was revived, and it was discovered that
Euclid's postulates are not categorical synthetic judgments, as Kant
maintained, 6 but simply assumptions. Such premises, it was found,
may be so freely and arbitrarily chosen that — subject to the condition
that they be mutually compatible — they may be allowed to contra-
dict the apparent evidence of the senses. Toward the close of the
century, as the result of the arithmetizing tendency in mathematical
analysis, it was further discovered that the concept of infinity, tran-
scending all intuition and analysis, could be introduced into mathe-
matics without impairing the logical consistency of the subject.
6 See Sammtliche Werke, II, passim.
Introduction 3
If the assumptions of mathematics are quite independent of the
world of the senses, and if its elements transcend all experience, 6
the subject is at best reduced to bare formal logic and at worst to
symbolical tautologies. The formal symbolic and arithmetizing tend-
ency in mathematics has met with remarkable success in the study
of the continuous. It has also led to stubborn paradoxes, a fact which
has aroused increased interest in the nature of mathematics: its scope
and place in intellectual life, the psychological source of its elements
and postulates, the logical force of its propositions and their validity
as interpretations of the world of sense perception.
The old idea that mathematics is the science of quantity, or of
space and number, has largely disappeared. The untutored intuition
of space, it is realized, leads to contradictions, a fact which upsets the
Kantian view of the postulates. Nevertheless the mathematician is
guided, although he is not controlled, by the external world of sense
perception. 7 The mathematical theory of continuity originated in
direct experience, but the definition of the continuum adopted in the
end by the mathematician transcends sensory imagination. From this,
mathematical formalists conclude that since we make no use of intui-
tion in mathematical definitions and premises, it is not necessary that
we should interpret the axioms or have any idea as to the nature of
the objects and relations involved. The intuitionists, however, insist
that the mathematical symbols involved should significantly express
thoughts. 8 Although there are two (or more) views of the grounds for
believing in the unassailable exactness of mathematical laws, the
recognition that mathematical concepts are suggested, although not
defined, by intuition thus easily accounts for the fact that the results
of mathematical deductive reasoning are in apparent agreement with
those of inductive experience. The derivative and the integral had
their sources in two of the most obvious aspects of nature — multi-
plicity and variability— but were in the end defined as mathematical
abstractions based on the fundamental concept of the limit of an
•Bertrand Russell has taken advantage of this disconcerting situation to define
mathematics facetiously as "the subject in which we never know what we are talking
about, nor whether what we are saying is true." "Recent Work on the Principles of
Mathematics," p. 84.
7 B6cher, "The Fundamental Concepts and Methods of Mathematics.
■ Brouwer, "Intuitionism and Formalism."
4 Introduction
infinite sequence of elements. Once we have traced this development,
the power and fecundity of these ideas when applied to the interpreta-
tion of nature will be easily understood.
The calculus had its origin in the logical difficulties encountered
by the ancient Greek mathematicians in their attempt to express
their intuitive ideas on the ratios or proportionalities of lines, which
they vaguely recognized as continuous, in terms of numbers, which they
regarded as discrete. It became involved almost immediately with
the logically unsatisfactory (but intuitively attractive) concept of the
infinitesimal. Greek rigor of thought, however, excluded the infinitely
small from geometrical demonstrations and substituted the circum-
ventive but cumbersome method of exhaustion. Problems of variation
were not attacked quantitatively by Greek scientists. No method
could be developed which would do for kinematics what the method
of exhaustion had done for geometry— indicate an escape from the
difficulties illustrated by the paradoxes of Zeno. The quantitative
study of variability, however, was undertaken in the fourteenth cen-
tury by the Scholastic philosophers. Their approach was largely
dialectical, but they had resort as well to graphical demonstration.
This method of study made possible in the seventeenth century the
introduction of analytic geometry and the systematic representation
of variable quantities.
The application of this new type of analysis, together with the free
use of the suggestive infinitesimal and the more extensive application
of numerical concepts, led within a short time to the algorithms of
Newton and Leibniz, which constitute the calculus. Even at this
stage, however, there was no clear conception of the logical basis of
the subject. The eighteenth century strove to find such a basis, and
although it met with little success in this respect, it did in the effort
largely free the calculus from intuitions of continuous motion and
geometrical magnitude. Early in the following century the concept
of the derivative was made fundamental, and with the rigorous defi-
nitions of number and of the continuum laid down in the latter half
of the century, a sound foundation was completed. Some twenty-five
hundred years of effort to explain a vague instinctive feeling for con-
tinuity culminated thus in precise concepts which are logically denned
but which represent extrapolations beyond the world of sensory ex-
Introduction 5
perience. Intuition, or the putative immediate cognition of an ele-
ment of experience which ostensibly fails of adequate expression, in
the end gave way, as the result of reflective investigation, to those
well-defined abstract mental constructs which science and mathematics
have found so valuable as aids to the economy of thought.
The fundamental definitions of the calculus, those of the derivative
and the integral, are now so clearly stated in textbooks on the subject,
and the operations involving them are so readily mastered, that it is
easy to forget the difficulty with which these basic concepts have been
developed. Frequently a clear and adequate understanding of the
fundamental notions underlying a branch of knowledge has been
achieved comparatively late in its development. This has never been
more aptly demonstrated than in the rise of the calculus. The precision
of statement and the facility of application which the rules of the cal-
culus early afforded were in a measure responsible for the fact that
mathematicians were insensible to the delicate subtleties required in
the logical development of the discipline. They sought to establish the
calculus in terms of the conceptions found in the traditional geometry
and algebra which had been developed from spatial intuition. During
the eighteenth century, however, the inherent difficulty of formulating
the underlying concepts became increasingly evident, and it then
became customary to speak of the "metaphysics of the calculus," thus
implying the inadequacy of mathematics to give a satisfactory exposi-
tion of the bases. With the clarification of the basic notions— which,
in the nineteenth century, was given in terms of precise mathematical
terminology — a safe course was steered between the intuition of the
concrete in nature (which may lurk in geometry and algebra) and the
mysticism of imaginative speculation (which may thrive on tran-
scendental metaphysics). The derivative has throughout its develop-
ment been thus precariously situated between the scientific phe-
nomenon of velocity and the philosophical noumenon of motion.
The history of the integral is similar. On the one hand, it has offered
ample opportunity for interpretations by positivistic thought in terms
either of approximations or of the compensation of errors — views
based on the admitted approximative nature of scientific measure-
ments and on the accepted doctrine of superimposed effects. On
the other hand, it has at the same time been regarded by idealistic
6 Introduction
metaphysics as a manifestation that beyond the finitism of sensory
percipiency there is a transcendent infinite which can be but asymp-
totically approached by human experience and reason. Only the pre-
cision of their mathematical definition — the work of the past century —
enables the derivative and the integral to maintain their autonomous
position as abstract concepts, perhaps derived from, but nevertheless
independent of, both physical description and metaphysical ex-
planation.
At this point it may not be undesirable to discuss these ideas, with
reference both to the intuitions and speculations from which they were
derived and to their final rigorous formulation. This may serve to
bring vividly to mind the precise character of the contemporary con-
ceptions of the derivative and the integral, and thus to make unam-
biguously clear the terminus ad quern of the whole development.
The derivative is the mathematical device used to represent point
properties of a curve or function. It thus has as its analogues in science
instantaneous properties of a body in motion, such as the velocity of
the object at any given time. When science is concerned with a time
interval, the average velocity over this interval is suitably defined as
the ratio of the change in the distance covered during the interval
to the time interval itself. This ratio is conveniently represented by
As
the notation — . Inasmuch as the laws of science are formulated by
At
induction on the basis of the evidence of the senses, on the face of it
there can be no such thing in science as an instantaneous velocity,
that is, one in which the distance and time intervals are zero. The
senses are unable to perceive, and science is consequently unable to
measure, any but actual changes in position and time. The power of
every sense organ is limited by a minimum of possible perception. 9
We cannot, therefore, speak of motion or velocity, in the sense of a
scientific observation, when either the distance or the corresponding
time interval becomes so small that the minimum of sensation in-
volved in its measurement is not excited — much less when the interval
is assumed to be zero.
If, on the other hand, the distance covered is regarded as a function
of the time elapsed, and if this relationship is represented mathe-
• Cf. Mach, Die Principien der WSrmelekre, pp. 71-77.
Introduction 7
matically by the equation 5 = /(/), the minimum of sensation no
longer operates against the consideration of the abstract difference
AS
quotient — . This has a mathematical meaning no matter how small
At
the time and distance intervals may be, provided, of course, that the
time interval is not zero. Mathematics knows no minimum interval
of continuous magnitudes — and distance and time may be considered
as such, inasmuch as there is no evidence which would lead one to
regard them otherwise. Attempts to supply a logical definition of such
an infinitesimal minimum which shall be consistent with the body of
mathematics as a whole have failed. Nevertheless, the term "instan-
taneous velocity" appears to imply that the time interval is to be
regarded not only as arbitrarily small but as actually zero. Thus the
term predicates the very case which mathematics is compelled to
exclude because of the impossibility of division by zero.
This difficulty has been resolved by the introduction of the deriva-
tive, a concept based on the idea of the limit. In considering the
As
successive values of the difference quotient — , mathematics may con-
At
tinue indefinitely to make the intervals as small as it pleases. In this
way an infinite sequence of values, r h r 2 , r 3 , . . . , r n , . . . (the suc-
As\
cessive values of the ratio — J is obtained. This sequence may be such
At)
that the smaller the intervals, the nearer the ratio r n will approach
to some fixed value L, and such that by taking the value of n to be
sufficiently large, the difference \L— r n \ can be made arbitrarily
small. If this be the case, this value L is said to be the limit of the
infinite sequence, or the derivative /'(/) of the distance function
f(i), or the instantaneous velocity of the body. It is to be borne in
mind, however, that this is not a velocity in the ordinary sense and
has no counterpart in the world of nature, in which there can be no
motion without a change of position. The instantaneous velocity as
thus defined is not the division of a time interval into a distance
ds
interval, howsoever much the conventional notation — = f'(t) may
at
suggest a ratio. This symbolism, although remarkably serviceable in
8 Introduction
the carrying out of the operations of the calculus, will be found to
have resulted from misapprehension on the part of Leibniz as to the
logical basis of the calculus.
The derivative is thus denned not in terms of the ordinary proc-
esses of algebra, but by an extension of these to include the concept
of the limit of an infinite sequence. Although science may not extra-
polate beyond experience in thus making the intervals indefinitely
small, and although such a process may be "inadequately adapted to
nature," 10 mathematics is at liberty to introduce the new limit con-
cept, on the basis of the logical definition given above. One can, of
course, make this notion still more precise by eliminating the words
"approach," "sufficiently large," and "arbitrarily small," as follows:
L is said to be the limit of the above sequence if, given any positive
number e (howsoever small), a positive integer N can be found
such that for n > N the inequality \L — r n \ < e is satisfied.
In this definition no attempt is made to determine any so-called
"end" of the infinite sequence, or to deal with the possibility that the
variable r n may "reach" its limit L. The number L, thus abstractly
defined as the derivative, is not to be regarded as an "ultimate ratio,"
nor may it be invoked as a means of "visualizing" an instantaneous
velocity or of explaining in a scientific or a metaphysical sense either
motion or the generation of continuous magnitudes. It is such unclear
considerations and unwarranted interpretations which, as we shall
see, have embroiled mathematicians, since the time of Zeno and his
paradoxes, in controversies which often misdirected their energy. On
the other hand, however, it is precisely such suggestive notions which
stimulated the investigations resulting in the formal elaboration of the
calculus, even though this very elaboration was in the end to exclude
them as logically irrelevant.
Just as the problem of defining instantaneous velocities in terms of
the approximation of average velocities was to lead to the definition
of the derivative, so that of defining lengths, areas, and volumes of
curvilinear configurations was to eventuate in the formulation of the
definite integral. This concept, however, was likewise ultimately to
be so defined that the geometrical intuition which gave it birth was
excluded. As part of the Qreek pursuit of unity in multiplicity, we
10 Schrodinger, Science and the Human Temperament, pp. 61-62.
Introduction 9
shall see that an attempt was made to inscribe successively within a
circle polygons of a greater and greater number of sides in the hope
of finally "exhausting" the area of the circle, that is, of securing a
polygon with so great a number of sides that its area would be equal
to that of the circle. This naive attempt was of course doomed to
failure. The same process, however, was adopted by mathematicians
as basic in the definition of the area of the circle as the limit A of the
infinite sequence formed by the areas A h At, A 3 , . . . , A n , ... of
the approximating polygons. This affords another example of extra-
polating beyond sensory intuition, inasmuch as there is no process
by which the transition from the sequence of polygonal areas to the
limiting area of the circle can be "visualized." An infinite subdivision
is of course excluded from the realm of sensory experience by the fact
that there exist thresholds of sensation. It must be banished also from
the sphere of thought, in the physiological sense, inasmuch as psychol-
ogy has shown that for an act of thought a measurable minimum of
duration of time is required. 11 Logical definition alone remains a
sufficient criterion for the validity of this limiting value A .
In order to free the limiting process just described from the geo-
metrical intuition inherent in the notion of area, mathematics was
constrained to give formal definition to a concept which should not
refer to the sense experience from which it had arisen. (This followed
a long period of indecision, the course of which we are shortly to
trace.) After the introduction of analytical geometry it became cus-
tomary, in order to find the area of a curvilinear figure, to substitute
for the series of approximating polygons a sequence of sums of approxi-
mating rectangles, as illustrated in the diagram (fig. 1). The area of
each of the rectangles could be represented by the notation /(#,) A*,-
n
and the sum of these by the symbolism S n = 2 /(*,-) A*,-. The area
of the figure could then be defined as the limit of the infinite sequence
of sums S n , as the number of subdivisions n increased indefinitely and
as the intervals Ax { approached zero. Having set up the area in this
manner with the help of the analytical representation of the curve, it
then became a simple matter to discard the geometrical intuition
leading to the formation of these sums and to define the definite
u Enriques, Problems of Science, p. 15.
10
Introduction
integral of /(#) over the interval from x = a to x = b arithmetically
n
as the limit of the infinite sequence of sums S n = 2 /(#.) A#; (where
the divisions A*,- are taken to cover the interval from a to b) as the
intervals Ax f become indefinitely small. This definition then invokes,
apart from the ordinary operations of arithmetic, only the concept of
the limit of an infinite sequence of terms, precisely as does that of the
derivative. The realization of this fact, however, followed only after
many centuries of investigation by mathematicians. The very nota-
tion i a f(x)dx, which is now customarily employed to represent this
definite integral, is again the result of the historical development of
the concept rather than of an effort to represent the final logical
FIGURE 1
formulation. It is suggestive of a sum, rather than of the limit of an
infinite sequence; and in this respect it is in better accord with the
views of Leibniz than with the intention of the modern definition.
The definite integral is thus defined independently of the derivative,
but Newton and Leibniz discovered the remarkable property con-
stituting what is commonly known as the fundamental theorem of the
calculus, viz., that the definite integral F(x) = Jlf(x)dx of the con-
tinuous function f(x) has a derivative which is this very same func-
tion, F'{x) = f(x). That is, the value of the definite integral of f(x)
from a to b may in general be found from the values, for x — a and
x = b, of the function F(x) of which f(x) is the derivative. This rela-
tionship between the derivative and the definite integral has been
called "the root idea of the whole of the differential and integral
Introduction 11
calculus." 12 The function F(x), when so denned, is often called the
indefinite integral of /(*), but it is to be recognized that it is in this
case not a numerical limit given by an infinite sequence, as is the
definite integral, but is a function of which f(x) is the derivative.
The function F(x) is sometimes also called the primitive of f{x),
and the value of F(b) — F(a) is occasionally taken as the definition
of Sif(x)dx. In this case the relationship ! b J(x)dx = Urn 2 /(*,) Ax, is
then the fundamental theorem of the calculus, rather than the defi-
nition of the definite integral.
Although the recognition of this striking inverse relationship,
together with the formulation of rules of procedure, may be taken as
constituting the invention of the subject, it is not to be supposed that
the inventors of the calculus were in possession of the above sophis-
ticated concepts of the derivative and the integral, so necessary in
the logical development of the new analysis. More than a hundred
years of investigation was to be required before the achievement of
their final definition in the nineteenth century.
It is the purpose of this essay to trace the development of these
two concepts from their incipiency in sense experience to their final
elaboration as mathematical abstractions, defined in terms of formal
logic by means of the idea of the limit of an infinite sequence. We
shall find that the history of the calculus affords an unusually striking
example of the slow formation of mathematical concepts by the
emancipation from all sense data of ideas born of our primary intui-
tions. The derivative and the integral are, in the last analysis, syn-
thetically defined in terms of ordinal considerations and not of those
of continuous quantity and variability. They are, nevertheless, the
results of attempts to schematize our sense impressions of these last
notions. This explains why the calculus, in the early stages of its de-
velopment, was bound up with concepts of geometry or motion, and
with explanations of indivisibles and the infinitely small; for these
ideas are suggested by naive intuition and experience of continuity.
There is in the calculus a further concept which merits brief con-
sideration at this point, not so much on account of any logical exigen-
cies in the present structure of the calculus as to make clearer its
historical development. The infinite sequences considered above in
u Courant, Differential and Integral Calculus, II, 111.
12 Introduction
the definitions of the derivative and of the integral were obtained by-
continuing, in thought, to diminish ad infinitum the intervals between
the values of the independent variable. Considerations which in
physical science led to the atomic theory were at various periods in the
development of the calculus adduced in mathematics. These made it
appear probable that just as in the actual subdivision of matter
(which has the appearance of being continuous) we arrive at ultimate
particles or atoms, so also in continuous mathematical magnitudes we
may expect (by means of successive subdivisions carried on in thought)
to obtain the smallest possible intervals or differentials. The derivative
would in this case be defined as the quotient of two such differentials,
and the integral would then be the sum of a number (perhaps finite,
perhaps infinite) of such differentials.
There is, to be sure, nothing intuitively unreasonable in such a
view; but the criterion of mathematical acceptability is logical self-
consistency, rather than reasonableness of conception. Such a view of
the nature of the differential, although possessing heuristic value in
the application of the calculus to problems in science, has been judged
inacceptable in mathematics because no satisfactory definition has as
yet been framed which is consistent with the principles of the calculus
as formulated above, or which may be made the basis of a logically
satisfactory alternative exposition. In order to retain the operational
facility which the differential point of view affords, the concept of the
differential has been logically defined, not in terms of mathematical
atomism, but as a notion derived from that of the derivative. The
differential dx of an independent variable x is to be thought of as
nothing but another independent variable; but the differential dy of
a function y = f(x) is defined as that variable the values of which
are so determined that for any given value of the variable dx the
. dy
ratio — shall be equal to the value of the derivative at the point in
dx
question, i. e., dy = f(x)dx. The differentials as thus defined are only
new variables, and not fixed infinitesimals, or indivisibles, or "ulti-
mate differences," or "quantities smaller than any given quantity,"
or "qualitative zeros," 13 or "ghosts of departed quantities," as they
u "Abominable little zeroes," they have been called by Osgood. See Osgood, "The
Calculus in Our Colleges and Technical Schools"; and Huntington, "Modern Interpreta-
tion of Differentials."
Introduction 13
have been variously considered in the development of the cal-
culus.
Poincare has said that had mathematicians been left the prey of
abstract logic, they would never have gotten beyond the theory of
numbers and the postulates of geometry. 14 It was nature which thrust
upon mathematicians the problems of the continuum and of the cal-
culus. It is therefore quite understandable that the persistent atom-
istic speculations of physical thought should have had a counterpart
in attempts to picture, by means of indivisible elements, the space
described by geometry. The further development of mathematics,
however, has shown that such notions must be abandoned, in order to
preserve the logical consistency of the subject. The basis of the con-
cepts leading to the derivative and the integral was first found in
geometry, for despite the apodictic character of its proofs, this sub-
ject was considered an abstract idealization of the world of the senses.
Recently, however, it has been more clearly perceived that mathe-
matics is the study of relationships in general and must not be ham-
pered by any preconceived notions, derived from sensory perception,
of what these relationships should be. The calculus has therefore been
gradually emancipated from geometry and has been made dependent,
through the definitions of the derivative and the integral, on the
notion of the natural numbers, an idea from which all traditional pure
mathematics, including geometry, can be derived. 15 Mathematicians
now feel that the theory of aggregates has provided the requisite
foundations for the calculus, for which men had sought since the time
of Newton and Leibniz. 16 It is impossible to predict with any con-
fidence, however, that this is the final step in the process of abstracting
from the primitive ideas of change and multiplicity all those irrelevant
incumbrances with which intuition binds these concepts. It is a
natural tendency of man to hypostatize those ideas which have great
value for him, 17 but a just appreciation of the origin of the derivative
and the integral will make clear how unwarrantedly sanguine is any
view which would regard the establishment of these notions as bringing
to its ultimate close the development of the concepts of the calculus.
14 See Osgood, op. cit., p. 457; Poincar6, Foundations of Science, p. 46.
16 Russell, Introduction to Mathematical Philosophy, p. 4. See also Poincare, Foundations
of Science, pp. 441, 462.
w Russell, The Principles of Mathematics, pp. 325-26.
17 Mach, The Science of Mechanics, p. 541.
II. Conceptions in Antiquity
THE PRE-HELLENIC peoples are usually regarded as pre-
scientific in their attitude toward nature, 1 inasmuch as they pal-
pably lacked the Greek confidence in its essential reasonableness, as
well as the associated feeling that beneath the perplexing hetero-
geneity and ceaseless flux of events would be found elements of
uniformity and permanence.
The search for universals, which the Greeks maintained so per-
sistently, apparently held no attraction for the Egyptians and the
Babylonians. So also the mathematical thought of these peoples —
about whom, among those of the ancient world, we are best informed —
bore no significant resemblance to ours, in that it lacked the tendency,
essential to both mathematical and scientific method, toward the
isolation and abstraction of certain samenesses from their confusingly
varied concomitants in nature and in thought. Lacking these ele-
ments of invariance to serve as premises of inference, they were
accordingly without appreciation of the characteristics which dis-
tinguish mathematics from science, namely, its logical nature and the
necessity of deductive proofs. 2
A large body of knowledge of spatial and numerical relations they
did, however, acquire; and the more familiar their work becomes, the
more it inspires our admiration. 3 It was, however, largely the result
of empirical investigations, or at best of generalizations which were
the result of incomplete induction from simple to more complicated
cases. The Egyptian rule for computing the volume of a square
pyramid — from which was obtained the most remarkable of all
1 On this point, however, there are significant differences of opinion. Barry (The
Scientific Habit of Thought, p. 104) places "the childhood of science" among the early
Greeks; Burnet (Greek Philosophy, Part I, "Thales to Plato," pp. 4-5) says "natural science
is the creation of the Greeks," and finds "not the slightest trace of that science in Egypt
or even in Babylon." On the other hand, Karpinski ("Is There Progress in Mathematical
Discovery," pp. 51-52) would regard the achievements of the Babylonians, Egyptians, and
Hindus as "scientific in the highest sense."
2 Milhaud, Nouvettes etudes sur I'histoire de la pens6e scientifique, pp. 41-133.
* See Neugebauer, Vorlesungen iiber Gesckichte der antiken mathematischen Wissenschaf-
ten, Vol. I, Vorgriechische Mathematik, for the best account of this work.
Conceptions in Antiquity 15
Egyptian results, the rule for determining the volume of a frustum
of a square pyramid — was probably the result of this method of pro-
cedure. 4 That the demonstration could not be correct, in our under-
standing of the mathematical implications of this term, is clear from
the fact that this result for the general case requires the use of infini-
tesimal or limit considerations 5 which, while constituting the point of
departure for the story of the derivative and integral, are not found
in any record before the Greek period. 6
More fundamental than this lack of deductive proofs of inferred
results is the fact that in all this Egyptian work the rules were applied
to concrete cases with definite numbers only. 7 There was no con-
ception in their geometry of a triangle as representative of all tri-
angles, 8 an abstract generalization necessary for the elaboration of a
deductive system. This lack of freedom and imagination is apparent
also in Egyptian arithmetic, into which the abstract number con-
cept, as such, did not enter, 9 and in which, with the exception of $,
all rational fractions were expressed as sums of unit fractions. 10
Babylonian mathematics resembles the Egyptian more than the
Greek, but with a stronger emphasis on the numerical side and thus
a more highly developed algebra than that of Egypt. Here again we
must not look for logical structure or proof, more complicated cases
being reduced to simpler, and so "proved," or rather treated as
analogous without proof; 11 and we must remember that this work, like
the Egyptian, deals with concrete cases only. In connection with
Babylonian astronomy, we find that problems involving continuous
variation were studied, but only to the extent of tabulating the values
of a function (such as the brightness of the moon, for example) for
* Ibid., p. 128.
5 See Dehn, "Ueber raumgleiche Polyeder," for proof that infinitesimal considerations
cannot here be avoided.
6 Neugebauer, op. cit., pp. 126-28. There seems to be no basis for the implication made
by Bell (The Search for Truth, p. 191) that the Egyptians used infinite or infinitesimal
considerations in deductive reasoning.
7 Neugebauer, op. cit., p. 127.
8 Luckey, "Was ist agyptische Geometrie?" p. 49.
9 Neugebauer, op. cit., p. 203. See also Miller, "Mathematical Weakness of the Early
Civilizations."
10 For example, the Egyptians did not regard as a single number that number which we
now represent by the symbol f , but thought of it as the sum of the two fractions, £ and fa
11 Neugebauer, op. cit., pp. 203-4.
16 Conceptions in Antiquity
values of the argument (time) measured at equal intervals, and from
this calculating the maximum (intensity) of the function. 12 The Greeks
were the first, however, systematically to analyze 13 the idea of contin-
uous magnitude and to develop concepts leading to the integral and
the derivative.
Our information on the history of mathematics in the interval be-
tween that of the best Egyptian and Babylonian mathematics and
the early work in Greece is unfortunately fragmentary. That these
oriental civilizations influenced Greek culture is clear; but the nature
and extent of their contribution is undetermined. However that may
be, there is an obvious change in spirit in both science and mathe-
matics, as these developed in Greece. The human mind was ''dis-
covered" as something different from the surrounding body of nature
and capable of discerning similarities in a multiplicity of events, of
abstracting these from their settings, generalizing them, and deducing
therefrom other relationships consistent with further experience. It is
for this reason that we consider mathematical and scientific method
as originating with the Hellenic race; 14 but to say that Greek mathe-
matics and science were autochthonous would be to forget the debt
of subject-matter owed to Egypt and Babylon. 15 It is likely 16 that the
new outlook of the Hellenes was the result of the flux of civilizations
occurring at this time, this impressing upon the rising Greek fortunes
the stamp of numerous cultures.
Thales is the first Greek mentioned in connection with this "in-
tellectual revolution," which produced elementary mathematics and
which was to reveal those difficulties in conception, the study and
resolution of which were to produce within the next twenty-five
hundred years the subject which we now call the calculus. He is said
to have been a great traveler, to have learned geometry from the
Egyptians and astronomy from the Babylonians, and upon his return
to Greece to have instructed his successors in the principles of these
u Hoppe, "Zur Geschichte der Infinitesimalrechnung bis Leibniz und Newton."
u Neugebauer, op. cit., p. 205.
" T. L. Heath, A History of Greek Mathematics, I, v, says, "Mathematics in short is a
Greek science, whatever new developments modern analysis has brought or may bring."
15 See Karpinski, "Is There Progress in Mathematical Discovery?" pp. 46-47. Cf. also
Gandz, "The Origin and Development of the Quadratic Equations in Babylonian, Greek
and Early Arabic Algebra," pp. 542-43.
14 As Neugebauer suggests (op. cit., p. 203).
Conceptions in Antiquity 17
subjects. Proclus says of his method of attack that it was "in some
cases more general, in other cases more empirical." Thales' demon-
strations may therefore have appealed to some extent to the evidence
of the senses, and in fact his theorems were those the truth of which
one would recognize by the execution of some practical construction. 17
To Thales, nevertheless, is ascribed the establishment of mathe-
matics as a deductive discipline. 18 He did not, however, construct a
body of mathematical knowledge, nor did he apply his method to the
analysis of the problem of the continuum. These tasks seem to have
been performed by Pythagoras, the second Greek mathematician of
whom we have substantial information. According to Proclus, he
"transformed the study of geometry into a liberal education, examin-
ing the principles of the science from the beginning and probing the
theorems in an immaterial and intellectual manner"; 19 but, beyond
admitting this, it is impossible to ascribe with any degree of certainty
other mathematical or scientific accomplishments to Pythagoras the
individual, since never in antiquity could he be distinguished from
his school and it is hardly possible to do so now. The knowledge
acquired by the school established by Pythagoras was held to be
strictly esoteric, with the result that when the general nature of
Pythagorean thought became apparent, after the death of the founder
about 500 b. c, it was already impossible to attribute to a single
member any one contribution. Nevertheless, the process of abstraction
begun by Thales was evidently completed by this school.
A new difficulty, however, then entered into Greek thought, for the
Pythagorean mathematical concepts, abstracted from sense impres-
sions of nature, were now in turn projected into nature and con-
sidered to be the structural elements of the universe. 20 Thus the
Pythagoreans attempted to construct the whole heaven out of numbers,
the stars being units which were material points. Later they identi-
fied the regular geometrical solids, with which they were familiar, with
the different sorts of substances in nature. 21 Geometry was regarded
17 Paul Tannery, La Gtomttrie grecque, pp. 89 ff.
» T. L. Heath, History of Greek Mathematics, I, 128.
19 Ibid., I, 140. Cf. also Moritz Cantor, Vorlesungen fiber Geschichte der Mathemaiik,
I, 137.
M See Brunschvicg, Les tiapes de la philosophic mathSmatique, pp. 34 ff.
n T. L. Heath, History of Greek Mathematics, I, 165.
18 Conceptions in Antiquity
by them as immanent in nature, and the idealized concepts of geom-
etry appeared to them to be realized in the material world. This
confusion of the abstract and the concrete, of rational conception and
empirical description, which was characteristic of the whole Pythag-
orean school and of much later thought, will be found to bear sig-
nificantly on the development of the concepts of the calculus. It has
often been inexactly described as mysticism, 22 but such stigmatization
appears to be somewhat unfair. Pythagorean deduction a priori
having met with remarkable success in its field, an attempt (unwar-
ranted, it is now recognized) was made to apply it to the description
of the world of events, in which Ionian hylozoistic interpretations a
posteriori had made very little headway. This attack on the problem
was highly rational and not entirely unsuccessful, even though it was
an inversion of the scientific procedure, in that it made induction
secondary to deduction.
One very important result of the Pythagorean search for unity in
nature and geometry was the theory of application of areas. This
originated with the Pythagoreans, if not with Pythagoras himself, 23
and became fundamental in Greek geometry, in which it later led to
the method of exhaustion, the Greek equivalent of our integration. 24
The method of the application of areas enabled them to say of a figure
bounded by straight lines that it was greater than, equivalent to, or
less than 25 a second figure. Such a superposition of one area upon
another constitutes the first step in the attempt to make exactly
definable the notion of area, in which a unit of area is said to be
contained in a second area a given number of times. Modern mathe-
matics has made fundamental the concept of number rather than that
of congruence, with the result that the word "area" no longer calls
vividly to mind that comparison of two surfaces which is essential in
this connection and which was always uppermost in Greek thought.
Greek mathematicians did not speak of the area of one figure, but of
the ratio of two surfaces, a definition which could not, because of the
problem of incommensurability, be made precise before a satisfactory
concept of number had been developed. Such a concept the Pythag-
22 See Russell, Our Knowledge of the External World, p. 19.
23 T. L. Heath, History of Greek Mathematics, I, 150.
24 It was basic also in the Greek solution of quadratic equations by geometrical algebra.
28 Our names for the conic sections (ellipse, hyperbola, and parabola) were, incidentally,
derived from the designations the Pythagoreans used in this connection.
Conceptions in Antiquity 19
oreans did not possess. This contribution was not made until the last
half of the nineteenth century, and it was to furnish, in the last
analysis, the basis of the whole of the calculus. However, to the Pythag-
oreans, in all probability, we owe the recognition of the need for some
such concept — a discovery which may be regarded as the first step, a
terminus a quo, in the development of the concepts of the calculus.
The inadequacy of the Pythagorean view of the ratio of magnitudes
was first made evident to the followers of this school on the application
of the doctrine, not to areas, but rather to the analogous comparison
of lines which is presupposed by our notion of length. Such investiga-
tions led the Pythagoreans to an intensely disconcerting discqvery.
If the side of a square were to be applied to the diagonal, no common
measure could be discovered which would express one in terms of the
other. In other words, these lines were shown to be incommensurable.
Just when this discovery took place and whether it was made by
Pythagoras himself, by the early Pythagoreans, or by later members
of the school are moot points in the history of mathematics. 26 It has
also been maintained that Pythagoras owed his knowledge of the
irrational and of the five regular solids, as well as much of his phi-
losophy, to the Hindus. 27
The question as to how the incommensurability was discovered or *
proved is also difficult to answer with any assurance. The rnethod of
application would suggest as a form of proof the geometrical equivalent
of the process of finding the greatest common divisor, but there is
another aspect of Pythagorean thought which points to a different sort
of reasoning. A prevailing belief in the unity and harmony of nature
and knowledge had led the Pythagoreans not only to explain different
aspects of nature by various mathematical abstractions, as already
suggested, but also to attempt to identify the realms of number and
magnitude. 28 By the term number, however, the Pythagoreans did
26 On this question see the following two papers by Heinrich Vogt: "Die Entdeckungs-
geschichte des Irrationalen nach Plato und anderen Quellen des 4. Jahrhunderts" : "Zur
Entdeckungsgeschichte des Irrationalen." Vogt concludes that the discovery was made by
the later Pythagoreans at some time before 410 b. c. T. L. Heath (History of Greek Mathe-
matics, I, 157) would place it "at a date appreciably earlier than that of Democritus. M
27 See Schroeder, Pythagoras und die Inder; and Vogt, "Haben die alten Inder den
Pythagoreischen Lehrsatz und das Irrationale gekannt?"
28 For a keenly critical account of the significance, from the scientific point of view, of
the Pythagorean problem of associating the fields of number and magnitude, see Barry,
The Scientific Habit of Thought, pp. 207 ff.
20 Conceptions in Antiquity
not understand the abstraction to which we give this name, but used
it to designate "a progression of multitude beginning from a unit and
a regression ending in it." 29 The integers were thus fundamental,
numbers being collections of units, and, as was the case with their
geometrical forms, they were immanent in nature, each having a posi-
tion and occupying a place in space. If geometrical abstractions were
the elements of actual things, number was the ultimate element of
these abstractions and thus of physical bodies and of all nature. 30
This hypostatization of number had led the Pythagoreans to regard a
line as made up of an integral number of units. This doctrine could not
be applied to the diagonal of a square, however, for no matter how
small a unit was chosen as a measure of the sides, the diagonal could
not be a "progression of multitude" beginning with this unit. The
proof of this fact, as given by Aristotle (and which possibly is that of
the Pythagoreans), 31 is based on the distinction between the odd and
the even, which the Pythagoreans themselves had emphasized.
The incommensurability of lines remained ever a stumbling block
for Greek geometry. That it made a strong impression on Greek
thought is indicated by the story, repeated by Proclus, that the
Pythagorean who disclosed the fact of incommensurability suffered
death by shipwreck as a result. It is demonstrated also, and more
reliably, by the prominence given to the doctrine of irrationals by
Plato and Euclid. It never occurred to the Greeks to invent an irra-
tional number 32 to circumvent the difficulty, although they did develop
as a part of geometry (found, for example, in the tenth book of
Euclid's Elements) the theory of irrational magnitudes. Failing to gen-
eralize their number system along the lines suggested later by the
development of mathematical analysis, the only escape for Greek
mathematicians in the end was to abandon the Pythagorean attempt
to identify the realm of number with that of geometry or of con-
tinuous magnitude.
The effort to unite the two fields was not given up, however, before
intuition had sought another way out of the difficulty. If there is no
» T. L. Heath, History of Greek Mathematics, I, 69-70.
30 Milhaud, Les Philosopkes geomitres de la Grece, p. 109.
31 Zeuthen, "Sur l'origine historique de la connaissance des quantites irrationelles."
32 Stolz, Vorlesungen fiber Allgemeine Arithmetik, I, 94; cf. also Vogt, Der Grenzbegriff in
der Elementar-mathematik, p. 48.
Conceptions in Antiquity 21
finite line segment so small that the diagonal and the side may both
be expressed in terms of it, may there not be a monad or unit of
such a nature that an indefinite number of them will be required for
the diagonal and for the side of the square?
We do not know definitely whether or not the Pythagoreans them-
selves invoked the infinitely small. We do know, however, that the
concept of the infinitesimal had entered into mathematical thought,
through a doctrine elaborated in the fifth century b. c, as the result
of Greek speculation concerning the nature of the physical world.
After the failure of the early Ionian attempts to find a fundamental
element out of which to construct all things, there arose at Abdera
the materialistic doctrine of physical atomism, according to which
there is no one physis, not even a small group of substances of which
everything is composed. The Abderitic school held that everything,
even mind and soul, is made up of atoms moving about in the void,
these atoms being hard indivisible particles, qualitatively alike but of
countless shapes and sizes, all too small to be perceived by sense
impressions.
There is nothing either logically or physically inconsistent in this
doctrine, which is a crude anticipation of our own chemical thought;
but the greatest of the Greek atomists, Democritus, did not stop
here: he was also a mathematician and carried the idea over into
geometry. As we now know from the Method of Archimedes, which was
discovered as a palimpsest in 1906, Democritus was the first Greek
mathematician to determine the volumes of the pyramid and the
cone. How he derived these results we do not know. The formula for
the volume of a square pyramid was probably known to the Egyp-
tians, 33 and Democritus in his travels may have learned of it and
generalized the result to include all polygonal pyramids. The result
for the cone would then be a natural inference from the result of
increasing indefinitely the number of sides in a regular polygon
forming the base of a pyramid. This explanation would correspond to
others involving similar infinitesimal conceptions, which we know
Democritus entertained and which later influenced Plato. 34
33 Neugebauer, Vorlesungen iiber Geschichte der Antiken mathematischen Wissenschaften,
p. 128.
34 See Luria, "Die Infinitesimaltheorie der antiken Atomisten."
22 Conceptions in Antiquity
Aristotle and Euclid ascribe to him a mathematical atomism, and
we know from Plutarch 35 that he was puzzled as to whether the
infinitesimal parallel circular sections of which the cone may be con-
sidered to be composed are equal or unequal: if they are equal, the
cone would be equal to the circumscribed cylinder; but if unequal,
they would be idented like steps. 36 We do not know how he resolved
this aporia, but it has been suggested that he made use of the idea of
infinitely thin circular laminae, or indivisibles, to find the volumes of
cones and cylinders, anticipating and using Cavalieri's theorem for
these special cases. 37 Democritus seems to have discriminated clearly
br tween physical and mathematical atoms — as did his later follower
Epicurus, although Aristotle made no such distinction 38 and according
to the much later account of Simplicius Democritus is said to have
held that all lines are divisible to infinity. 39 However, since most of
Democritus' work is lost, we cannot now reconstruct his thought.
That he was interested in other mathematical problems bearing on
the infinitesimal we know from the titles of works now lost, but
which are referred to by Diogenes Laertius. One of these seems to
have been on horn angles (the angles formed by curves which have a
common tangent at a point), and another on irrational (incommen-
surable) lines and solids. 40 It may be inferred, therefore, that the
Pythagorean difficulty with the incommensurable was probably
familiar to him, and it may be that he tried to solve it by some theory
of mathematical atomism. It has been maintained 41 that Democritus
was too good a mathematician to have had anything to do with such
a theory as that of indivisible lines; but it is difficult to imagine
how a mathematical atom is to be conceived if not as an indivisible.
At all events, whatever his conception of the nature of infinitesimals
may have been, the influence of Democritus has persisted. The idea
of the fixed infinitesimal magnitude has clung tenaciously to mathe-
matics, frequently to be invoked by intuition when logic apparently
failed to offer a solution, and finally to be displaced in the last century
by the rigorous concepts of the derivative and the integral.
35 Plutarch, Miscellanies and Essays, IV, 414-16.
34 See also T. L. Heath, History of Greek Mathematics, I, 180; and Luria, op. cit., pp.
138-40, for statements of this "paradox."
" Simon, Geschichte der Mathematik im Altertum, p. 181. *• Luria, op. cit., pp. 179-80.
M Cf. Simplicii commentarii in octo Aristotelis physicae auscultationis libros, p. 7.
« T. L. Heath, History of Greek Mathematics, 1, 179-81. « Ibid., p. 181.
Conceptions in Antiquity 23
That the infinitesimal was not eagerly welcomed into Greek geom-
etry after the time of the Pythagoreans and Democritus may have
been due largely to a school of philosophy that had risen at Elea, in
Magna Graecia. The Eleatic school, although not essentially mathe-
matical, was apparently familiar with, and probably influenced by,
Pythagorean mathematical philosophy; but it became an opponent of
the chief tenet of this thought. Instead of proclaiming the constitution
of objects as an aggregate of units, it pointed out the apparent con-
tradictions inherent in such a doctrine, maintaining against the atomic
view the essential oneness and changelessness of the world. This
stultifying monism was upheld by Parmenides, the leader of the
school, with perhaps a touch of skepticism derived from his iconoclastic
predecessor, the poet-philosopher Xenophanes.
In an indirect defense of this doctrine, the Eleatics proceeded to
demolish, with skillful dialectic, the basis of opposing schools of
thought. The most damaging arguments were offered by Zeno, the
student of Parmenides. After presenting the obvious objection to the
Pythagorean indefinitely small monad — that if it has any length, an
infinite number will constitute a line of infinite length; and if it has
no length, then an infinite number will likewise have no length — he
added the following general dictum against infinitesimals: "That
which, being added to another does not make it greater, and being
taken away from another does not make it less, is nothing." 42 More
critical and subtle than these, however, are his four famous paradoxes
on motion. 43 There has been much speculation as to the purpose of
Zeno's arguments, 44 lack of evidence making it impossible to decide
conclusively against whom they were directed: whether against the
Pythagoreans, or the atomists, or Heraclitus, or whether they were
mere sophisms. That they were intended merely as dialectical puzzles
may perhaps be indicated by the passage in Plutarch's life of Pericles:
Also the two-edged tongue of mighty Zeno, who,
Say whp.t one would, could argue it untrue. 45
** Zeller, Die Philosopkie der Griechen in ihrer Geschichtlichen Entwicklung, I, 540.
*• For an unusually extensive account of the history of Zeno's paradoxes, with bibli-
ography, see Cajori's article on "History of Zeno's Arguments on Motion."
** On this subject see Cajori, "The Purpose of Zeno's Arguments on Motion." This
article includes an account of the varying interpretations, as well as extensive biblio-
graphical notes.
46 Plutarch, The Lives of the Noble Grecians and Romans, p. 185.
24 Conceptions in Antiquity
On the other hand there is some reason to suppose that these
arguments were presented in connection with a more significant pur-
pose. It is not improbable that Zeno, although he was neither a
mathematician nor a physicist, propounded the paradoxes to point
out the weakness in the Pythagorean definition of a point as unity
having position, and in the resulting Pythagorean multiplicity which
did not distinguish clearly betw r een the geometrical and the physical. 46
Pythagorean science and mathematics had been concerned with
form and structure, and not with flux and variability; but had the
Pythagoreans applied their philosophy to the aspects of change in
nature rather than to those of permanence, the resulting explanation
of motion would have been in terms of concepts attacked by Zeno in
his third and fourth paradoxes (those of the arrow and the stade), 47 in
which space and time are assumed to be composed of indivisible
elements. The arguments would hold equally well, of course, against
mathematical atomism. The first two paradoxes (the dichotomy and
the Achilles 46 ) are directed against the opposite conception, that of the
infinite divisibility of space and time, and are based upon the impos-
sibility of conceiving intuitively the limit of the sum of an infinite
series. The four paradoxes are, of course, easily answered in terms of
46 See Paul Tannery, La GeomUrie grecque, p. 124; cf . also the same author, "Le Concept
scientifique du continu. Zenon d'Elee et Georg Cantor." Milhaud (Nouvelles ttudes, pp.
153-54) and Cajori concur in the view here presented. See further, Cajori, "Purpose of
Zeno's Arguments."
47 The argument in the arrow is as follows: Anything occupying space equal to itself (or
in one and the same place) is at rest; but this is true of the arrow at every moment of its
flight. Therefore the arrow does not move. (See The Works of Aristotle, Vol. II, Physica VI.
239b, for the statement of the paradoxes.) The argument in the stade, as given by Aristotle,
is obscure (because of brevity), but is equivalent to the following: Space and time being
assumed to be made up of points and instants, let there be given three parallel rows of
points, A, B, and C. Let C move to the right and A to the left at the rate of one point per
instant, both relative to B; but then each point of A will move past two points of C in an
instant, so that we can subdivide this, the smallest interval of time; and this process can
be continued ad infinitum, so that time can not be made up of instants.
48 The argument in the dichotomy is as follows: before an object can traverse a given
distance, it must first traverse half of this distance; before it can cover half, however, it
must cover one quarter; and so ad infinitum. Therefore, since the regression is infinite,
motion is impossible, inasmuch as the body would have to traverse an infinite number of
divisions in a finite time. The argument in the Achilles is similar. Assume a tortoise to have
started a given distance ahead of Achilles in a race. Then by the time Achilles has reached
the starting point of the tortoise, the latter will have covered a certain distance; in the time
required by Achilles to cover this additional distance, the tortoise will have gone a little
farther; and so ad infinitum. Since this series of distances is infinite, Achilles can never
overtake the tortoise, for the same reason as that adduced in the dichotomy.
Conceptions in Antiquity 25
the concepts of the differential calculus. There is no logical difficulty
in the dichotomy or the Achilles, the uneasiness being due merely to
failure of the imagination to realize, in terms of sense impressions, the
nature of infinite convergent series which are fundamental in the
precise explanation of, but not involved in our obscure notion of,
continuity. The paradox of the flying arrow involves directly the
conception of the derivative and is answered immediately in terms of
this. The argument in this paradox, as also that in the stade, is met
by the assumption that the distance and time intervals contain an
infinite number of subdivisions. Mathematical analysis has shown that
the conception of an infinite class is not self-contradictory, and that
the difficulties here, as also in the case of the first two paradoxes, are
those of conceiving intuitively the nature of the continuum and of
infinite aggregates. 49
In a broad sense there are no insoluble problems, but only those
which, arising from a vague feeling, are not yet suitably expressed. 60
This was the position of Zeno's paradoxes in Greek thought; for the
notions involved were not given the precision of expression necessary
for the resolution of the putative difficulties. It is clear that the
answers to Zeno's paradoxes involve the notions of continuity, limits,
and infinite aggregates — abstractions (all related to that of number)
to which the Greeks had not risen and to which they were in fact
destined never to rise, although we shall see Plato and Archimedes
occasionally straining toward such views. That they did not do so
may have been the result of their failure, indicated above in the case
of the Pythagoreans, clearly to separate the worlds of sense and
reason, of intuition and logic. Thus mathematics, instead of being the
science of possible relations, was to them the study of situations
thought to subsist in nature.
The inability of Greek mathematicians to answer in a clear manner
the paradoxes of Zeno made it necessary for them to forego the
attempt to give to the phenomena of motion and variability a quanti-
tative explanation. These experiences were consequently confined to
the field of metaphysical speculation, as in the work of Heraclitus, or
* Accounts of the mathematical resolutions of Zeno's paradoxes are given in the works
of Bertrand Russell, The Principles of Mathematics and Our Knowledge of the External
World.
80 Enriques, Problems of Science, p. 5.
36 Conceptions in Antiquity
to that of qualitative description, as the physics of Aristotle. Only the
static aspects of optics, mechanics, and astronomy found a place in
Greek mathematics, and it remained for the Scholastics and early
modern scientists to establish a quantitative dynamics. Zeno's argu-
ments and the difficulty of incommensurability had also a more
general effect on mathematics: in order to retain logical precision, it
was necessary to give up the abortive Pythagorean effort to identify
the domains of number and geometry, and to abandon also the pre-
mature Democritean attempt to explain the continuous in terms of
the discrete. It is, however, impossible satisfactorily to interpret the
world of nature and the realm of geometry (spheres which for the
Greeks were not essentially distinct) without superimposing upon
them a framework of discrete multiplicity; without ordering, by means
of number, the heterogeneity of impressions received by the senses;
and without at every point comparing nonidentical elements. Thought
itself is possible only in terms of a plurality of elements. As a conse-
quence, the concept of discreteness cannot be excluded completely
from the study of geometry. The continuous is to be interpreted in
terms of successive subdivision, that is to say, in terms of the discrete,
although from the Greek point of view the former could not be logically
identified with the latter. The clever manner in which the method of
successive subdivision was applied in Greek geometry, without the
loss of logical rigor, will be seen later in the method of exhaustion — a
procedure which was developed, not in Italy, but in and around the
Greek mainland, whither many Pythagoreans wandered, on the
breaking up of the school, toward the beginning of the fifth century
b. c. Zeno likewise lived for a time in Athens, the rising center of
Greek culture and mathematics. Here Pericles, the political leader
of that city in its Golden Age, is said to have been one of his
listeners. 81
At Athens the great philosopher Plato, although himself not pri-
marily a mathematician, was conversant with, and displayed a lively
interest in, the problems of the geometers. He may not have contrib-
uted much original work in mathematics, but he advanced the sub-
ject, nevertheless, through his great enthusiasm for it. He is said to
have paid particular attention to the principles of geometry — to the
" Plutarch, Lives, p. 185.
Conceptions in Antiquity 27
hypotheses, definitions, methods. 62 For this reason he was particularly
concerned with the difficulties which led eventually to the calculus.
In his dialogues he considered the Pythagorean problem of the nature
of number and its relationship to geometry, 53 the difficulty of incom-
mensurability, 64 the paradoxes of Zeno, 66 and the Democritean ques-
tion of indivisibles and the nature of the continuum. 56
Plato seems to have realized the gulf between arithmetic and
geometry, and it has been conjectured 67 that he may have tried to
bridge it by his concept of number and by the establishment of arith-
metic upon a firm axiomatic basis similar to that which was built up
in the nineteenth century independently of geometry; but we cannot
be sure, because these thoughts do not occur in his exoteric writings
and were not advanced by his successors. If Plato made an attempt
to arithmetize mathematics in this sense, he was the last of the
ancients to do so, and the problem remained for modern mathe-
matical analysis to solve. The thought of Aristotle we shall find
diametrically opposed to any such conceptions. It has been suggested
that Plato's thought was so opposed by the polemic of Aristotle that
it was not even mentioned by Euclid. Certain it is that in Euclid
there is no indication of such a view of the relation of arithmetic to
geometry; but the evidence is insufficient to warrant the assertion 68
that, in this connection, it was the authority of Aristotle which held
back for two thousand years a transformation which the Academy
sought to complete. A sound basis for either mechanics or arithmetic
must be built upon the limit concept — a notion which is not found in
the extant works of Plato nor in those of his successors. The Platonists,
on the contrary, attempted to develop the misleading idea of indi-
visibles or fixed infinitesimals, a notion which the modern arith-
metization of analysis has had cause to reject.
H To Plato are ascribed, among other things, the formulation of the analytic method
and the restriction of Euclidean geometry to constructions possible with ruler and compass
only. Hankel (Zur Gesckichte der Mathematik in Alterthum und Mittelalter, p. 156) ascribes
this limitation to Plato, but T. L. Heath (History of Greek Mathematics, I, 288) would
place it earlier.
63 Republic VII. 525-27.
M See, in particular, Theaetetus 147-48; Laws 819d-820c.
66 Parmenides 128 ff.
68 Philebas 17 ff.
67 Toeplitz, "Das Verhaltnis von Mathematik und Ideenlehre bei Plato."
68 Ibid., pp. 10-11.
28 Conceptions in Antiquity
Plato apparently did not give direct answers to the difficulties
involved in incommensurability or in Zeno's paradoxes, although he
expressed his opposition to the Pythagorean concepts of infinity, and
of the monad as being unity having position, 69 and also to Democritean
atomism. He was strongly influenced by both of these schools, but
apparently felt that their views were too much the result of sense
experience. Plato's criterion of reality was not consistency in experi-
ence but reasonableness in thought. For him, as for the Pythagoreans,
there was no necessary distinction between mathematics and science;
both were the result of deduction from clearly perceived first principles.
The Pythagorean monad and the Democritean mathematical atom-
ism,, which gave every line a thickness, perhaps appealed too strongly
to materialistic sense experience to suit Plato, so that he had recourse
to the highly abstract apeiron or unbounded indeterminate. This was
the eternally moving infinite of the Ionian philosopher, Anaximander, 60
who had suggested it in opposition to Thales' less subtle assertion that
the concrete material element, water, was the basis of all things.
According to Plato, the continuum, could better be regarded as gen-
erated by the flowing of the apeiron than thought of as consisting of
an aggregation (however large) of indivisibles. This view represents a
fusion of the continuous and the discrete not unlike the modern
intuitionism of Brouwer. 61 The infinitely small was apparently not to
be reached through a continued subdivision, 62 but was to be regarded,
perhaps, as analogous to the generative infinitesimal of Leibniz, or
the "intensive" infinitely small magnitude which appeared in ideal-
istic philosophy in the nineteenth century. Mathematics has found it
necessary to discard both views in making the infinitesimal subor-
dinate to the derivative in the logical foundation of the calculus.
However, the notion of the infinitesimal proved very suggestive in the
early establishment of the calculus, and, as Newton remarked more
than two thousand years later, the application to it of our intuitions
of motion removes from the doctrine much of the harshness felt in
the mathematical atomism of Democritus and later of Cavalieri.
This, however, necessarily led to a loss both of precise logical defi-
69 See T. L. Heath, History of Greek Matliematics, I, 293.
60 Hoppe. "Zur Geschichte der Infinitesimalrechnung," p. 154; see also Milhaud, Les
Philosophes giometres, p. 68.
81 Helmholtz, Counting and Measuring, pp. xxii-xxiv. 82 Hoppe, op. cit., p. 152.
Conceptions in Antiquity 29
nition and of clear sensory interpretation, neither of which Plato
supplied. 63
The belief that mathematics becomes "sterilized by losing contact
with the world's work" 64 is widely held but is not easily justified.
The conjunction of mathematics and philosophy, as found, for ex-
ample, in Plato, Descartes, and Leibniz, has been perhaps as valuable
in suggesting new advances as has the blending of mathematical and
scientific thought illustrated by Archimedes, Galileo, and Newton.
The disregard in Platonic thought of any basis in the evidences of
sense experience has not unjustly been regarded, from the scientific
point of view, as an "unmitigated misfortune." On the other hand,
the successful development of his views would have given to mathe-
matics — which is interested solely in relationships which are logically
thinkable rather than in those believed to be realized in nature — a
flexibility and an independence of the world of sense impressions
which were to be essential for the ultimate formulation of the con-
cepts of the calculus. One may therefore say, in a very general sense,
that "we know from Plato's own writings that he was thinking out
the solution of problems that lead directly to the discovery of the
calculus." 66 It is, however, altogether too much to assert that "Indeed
there are probably only four or five names of mathematical dis-
coverers that stand between Plato on the one hand and Newton and
Leibniz, the discoverers of the calculus, on the other hand." 66 We
shall see that the calculus was the result of a long train of mathe-
matical thought, developed slowly and with great difficulty by very
many thinkers.
That the doctrines of the continuous and the infinitesimal did not
develop along the abstract lines vaguely indicated by Plato was
probably the result of the fact that Greek mathematics included no
general concept of number, 67 and, consequently, no notion of a con-
tinuous algebraic variable upon which such theories could logically
have been based. Disregard of the abstract idealizations which Plato
83 Hoppe {op. cit., p. 152) asserts that in Plato one finds the first clear conception of the
infinitesimal. It is difficult, however, to perceive on what grounds such a thesis is to be
defended.
84 Hogben, Science for the Citizen, p. 64.
66 Marvin, The History of European Philosophy, p. 142.
M Ibid.
67 Miller, "Mathematical Weakness of Early Civilizations."
S6 Conceptions in Antiquity-
suggested, but never clearly defined, may also have been due in some
measure to the opposition afforded by the inductive scientific views of
Aristotle and the Peripatetic School. Aristotelian thought, while not
destroying the rigorously deductive character of Greek geometry, may
have preserved in Greek mathematics that strong reasonable and
matter-of-fact cast which one finds in Euclid and which operated
against the early development of the calculus, as well as against the
Platonic tendency toward speculative metaphysics.
Although Plato did not solve the difficulties which the Pythagoreans
and Democritus encountered, he urged their study upon his asso-
ciates, inveighing against the ignorance concerning such problems
which prevailed among the Greeks. 68 To Eudoxus he is said to have
proposed a number of problems in stereometry which proved to be
remarkably suggestive in leading toward the calculus. In this con-
nection the demonstrations which Eudoxus gave of the propositions
(previously stated without proof by Democritus) on the volumes of
pyramids and cones led to his famous general method of exhaustion
and to his definition of proportion. The achievements of Eudoxus are
those of a mathematician who was at the same time a scientist, with
none of the occult or mystic in him. 69 As a consequence, they are based
at every point on finite, intuitively clear, and logically precise consid-
erations. In method and spirit the later work of Euclid will be found
to owe much more to Eudoxus than to Plato.
We have seen that the Pythagorean theory of proportion could not
be applied to all lines, many of which are incommensurable, and
that the Democritean view of infinitesimals was logically untenable.
Eudoxus proposed means by which these difficulties could be avoided.
The paths he indicated, in his theory of proportion and in the method
of exhaustion, were not the equivalents of our modern conceptions of
number and limit, but rather detours which obviated the necessity of
using the latter. They were significant, however, in that they made it
possible for the Greek mind confidently to pursue its attack upon prob-
lems which were to eventuate much later in the calculus.
The Pythagorean conception of proportion had been the result of
the identification of geometrical magnitudes and integral numbers.
M See Laws 819d-820c.
•» T. L. Heath, History of Greek Mathematics, I, 323-25; Becker, "Eudoxos-Studien,"
1936, p. 410.
Conceptions in Antiquity 31
Two lines, for example, were to each other as the ratio of the (integral)
numbers of units in each. With the discovery of the incommensura-
bility of some lines with others, however, this definition could no longer
be universally applied. Eudoxus substituted for it another which was
more general, in that it did not require two of the terms in the propor-
tion to be (integral) numbers, but allowed all four to be geometrical
entities, and required no extension of the Pythagorean idea of number.
Euclid 70 states Eudoxus' definition as follows: "Magnitudes are said
to be in the same ratio, the first to the second and the third to the
fourth, when, if any equimultiples whatever be taken of the first
and third, and any equimultiples whatever of the second and fourth,
the former equimultiples alike exceed, are alike equal to, or alike fall
short of, the latter equimultiples respectively taken in corresponding
order." 71 The theory of proportion thus stated involves only geo-
metrical quantities and integral multiples of them, so that no general
definition of number, rational or irrational, is necessary. It is interesting
to see that after the development of mathematical analysis, the con-
cept of proportion resembles the arithmetical form of the Pythag-
oreans rather than the geometrical one of Eudoxus. Even when the
ratio is not expressible as the quotient of two integers, we now sub-
stitute for it a single number and symbol such as v or e. Although
Eudoxus did not, as we do, regard the ratio of two incommensurable
quantities as a number, 72 nevertheless his definition of proportion ex-
presses the ordinal idea involved in the present conception of real
number. The assertion that it is "word for word the same as the
general definition of number given by Weierstrass" 73 will be found,
however, to be incorrect, both literally and in its implications. The
formulation of Eudoxus was, on the contrary, a means of avoiding the
need of such an arithmetic definition as that of Weierstrass.
The method of exhaustion of Eudoxus shows the same abandon-
ment of numerical conceptions which we have seen in his theory of
proportion. Length, area, and volume are now carefully defined
numerical entities in mathematics. After the time of the Pythagoreans,
classic Greek mathematics did not attempt to identify number with
70 Book V, Definition 5.
n The Thirteen Books of Euclid's Elements, trans, by T. L. Heath, II, 114.
n Stolz, Vorlesungen iiber aUgemeine Arithmetik, I, 94.
71 Simon, "Historische Bemerkungen iiber das Continuum," p. 387.
32 Conceptions in Antiquity
geometrical quantities. As a result no rigorous general definitions of
length, area, and volume could then be given, the meaning of these
quantities being tacitly understood as known from intuition. The
question, "What is the area of a circle?" would have had no meaning
to the Greek geometers. But the query, "What is the ratio of the
areas of two circles?" would have been a legitimate one, and the
answer would have been expressed geometrically: "the same as that
of squares constructed on the diameters of the circles." 74 The fact that
squares and circles are incommensurable with each other does not
cause any incongruity in the idea of their entering into the same pro-
portion under the general definition of Eudoxus; but the proof of the
correctness of the proportion requires in this case the comparison of
squares with squares and of circles with circles.
Obviously the old Pythagorean method of the application of areas
cannot be employed in the case of circles, so Eudoxus had recourse
to an idea which had been advanced sometime before by Antiphon
the Sophist and again a generation later by Bryson. These men had
inscribed within a circle a regular polygon, and by successively
doubling the number of sides they seem to have hoped to reach a
polygon which would coincide with the circle and so "exhaust" its
area. It should, however, be borne in mind that we do not know just
what Antiphon (and later Bryson) said. The method of Antiphon has
been described 75 as equivalent at one and the same time to the method
of Eudoxus (as given in Euclid XII, 2), and to our conception of the
circle as the limit of such an inscribed polygon, but merely expressed
in different terminology. This cannot be strictly correct. If Antiphon
had considered the process of bisection as carried out to an infinite
number of steps, he would not have been thinking in the terms of
Eudoxus and Euclid, as we shall see. If, on the other hand, he did not
regard the process as continued indefinitely but only as carried out to
any desired degree of approximation, he could not have had our idea
of a limit. Furthermore, our conception of the limit is numerical,
whereas the notions of Antiphon and Eudoxus are purely geometrical.
The suggestive idea of Antiphon, however, was adopted by Bryson,
who is reputed not only to have inscribed a polygon within the circle
74 Cf. Vogt, Der Grenzbegrijf in der Elementar-mathematik, p. 42.
78 T. L. Heath, History of Greek Mathematics, I, 222.
Conceptions in Antiquity 33
but also to have circumscribed one about it as well, saying that the
circle would ultimately, as the result of continued bisection, be the
mean of the inscribed and circumscribed polygons. Again we do not
know exactly what he said, and cannot tell clearly what he meant. 76
Interpretations have been advanced 77 which would go so far as to see
in the work of Bryson the concept of a "Dedekind Cut" or of the con-
tinuum of Georg Cantor, but the evidence would hardly appear to
warrant such imputations. However, the idea which he suggested was
developed by Eudoxus into a rigorous tyDe of argument for dealing
with problems involving two dissimilar, heterogeneous, or incom-
mensurable quantities, in which intuition fails to represent clearly the
transition from one to the other which is necessary to make a com-
parison possible.
The procedure which Eudoxus proposed has since become known
as the method of exhaustion. The principle upon which this method
is based is commonly called the lemma, or postulate, of Archimedes,
although the great Syracusan mathematician himself ascribed it 78 to
Eudoxus and it is not improbable that it had been formulated still
earlier by Hippocrates of Chios. 79 This axiom (as given in Euclid X,
1) states that, given two unequal magnitudes (neither equal to zero,
of course, since for the Greeks this was neither a number nor a mag-
nitude), "if from the greater there be subtracted a magnitude greater
than its half, and from that which is left a magnitude greater its half,
and if this process be repeated continually, there will be left some
magnitude which will be less than the lesser magnitude set out." 80
This definition (in which, of course, any ratio may be substituted in
place of one-half) excluded the infinitesimal from all demonstrations
in the geometry of the Greeks, although we shall find this banished
notion entering occasionally into their thought as an explorative aid.
From the fact that, on continuing the process indicated in the axiom
of Archimedes, the magnitude remaining can be made as small as we
please, the procedure introduced by Eudoxus came much later to be
74 Ibid., I, 224.
77 See Becker, "Eudoxos-Studien," in particular, 1933, pp. 373-74; Toeplitz, "Das Ver-
haltnis von Mathematik und Ideenlehre bei Plato," pp. 31-33.
78 In his Quadrature of the Parabola. See T. L. Heath, History of Greek Mathematics, I,
327-28.
79 Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, p. 122.
80 Euclid, Elements, Heath trans., vol. HI, p. 14.
34 Conceptions in Antiquity
called the method of exhaustion. It is to be remarked, however, that
the word exhaustion was not applied in this connection until the
seventeenth century, 81 when mathematicians somewhat ambiguously
and uncritically employed the term indifferently to designate both the
ancient Greek procedure and their own newer methods which led
immediately to the calculus and which truly "exhausted" the mag-
nitudes.
The Greek mathematicians, however, never considered the process
as being literally carried out to an infinite number of steps, as we do
in passing to the limit — a concept which allows us to interpret the
area or volume as truly exhausted, or at least as defined as the limit
of the infinite numerical sequence obtained in this manner. There
was always, in the Greek mind, a quantity left over (although this
could be made as small as desired), so that the process never passed
beyond clear intuitional comprehension. A simple illustration will per-
haps serve to make the nature of the method clear. The proposition,
in Euclid XII, 2, that the areas of circles are to each other as the
squares on their diameters will suffice for this purpose. The substance
of this proof is as follows: Let the areas of the circles be A and a, and
let their diameters be D and d respectively. If the proportion a : A
= d 2 :D 2 is not true, then let a' : A = d 2 : D 2 , where a' is the area
of another circle either greater or smaller than a. If a' is smaller
than a, then in the circle of area a we can inscribe a polygon of area p
such that p is greater than a' and smaller than a. This follows from
the principle of exhaustion (Euclid X, 1) — that if from a magnitude
(such as the difference in area between a' and a) we take more than
its half, and from the difference more than its half, and so on, the
difference can be made less than any assignable magnitude. If P is
the area of a similar polygon inscribed in the circle of area A, then
we know that p : P — d 2 : D 2 = a' :A. But since p > a', then P >
A, which is absurd, since the polygon is inscribed within the circle.
In a similar manner it can be shown that the supposition a! > a like-
wise leads to a contradiction, and the truth of the proposition is
therefore established. 82
The method of exhaustion, although equivalent in many respects
81 In particular in Gregory of St. Vincent, Opus geometricum, pp. 739-40,
82 Cf. Euclid, Elements, Heath trans., vol. Ill, pp. 371-78.
Conceptions in Antiquity 35
to the type of argument now employed in proving the existence of a
limit in the differential and the integral calculus, does not represent
the point of view involved in the passage to the limit. The Greek method
of exhaustion, dealing as it did with continuous magnitude, was
wholly geometrical, for there was at the time no knowledge of an
arithmetical continuum. This being the case, it was of necessity based
on notions of the continuity of space — intuitions which denied any
ultimate indivisible portion of space, or any limit to the divisibility
in thought of any line segment. The inscribed polygon could be made
to approach the circle as nearly as desired, but it could never become
the circle, for this would imply an end in the process of subdividing
the sides. However, under the method of exhaustion it was not neces-
sary that the two should ever coincide. By an argument based upon
the reductio ad absurdum, it could be shown that a ratio greater or
less than that of equality was inconsistent with the principle that the
difference could be made as small as desired.
The argument of Eudoxus appealed at every stage to intuitions of
space, and the process of subdivision made no use of such unclear con-
ceptions as that of a polygon with an infinite number of sides — that
is, of a polygon which should ultimately coincide with the circle.
No new concepts were involved, and the gap between the curvilinear
and the rectilinear still remained unspanned by intuition. Eudoxus,
however, had most ingeniously contrived to demonstrate — without
resort to the logically self -contradictory infinitesimal previously
invoked by vague imagination — the truth of certain geometrical
propositions requiring a comparison of the curvilinear with the
rectilinear and of the irrational with the rational.
There is no logical difficulty to be found in the argument used in
the method of exhaustion, but the cumbersomeness of its application
led later mathematicians to seek a more direct approach to problems
in which the application of some such procedure would have been
indicated. The method of exhaustion has, most misleadingly, been
characterized as "a well-established algorithm of the differential cal-
culus." 83 It is indeed true that the problems to which the method was
applied were those which led toward the calculus. Nevertheless, it is
not incorrect to say that the procedure involved actually directed
83 Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 116.
36 Conceptions in Antiquity-
attention away from the discovery of an equivalent algorithm, in that
it directed attention toward the synthetic form of exposition rather
than toward an analytic instrument of discovery. 84 It did represent
a conventional type of demonstration, but the Greek mathematicians
never developed this into a concise and well-recognized operation with
a characteristic notation. In fact the ancients never made the first
step in this direction: they did not formulate the principle of the
method as a general proposition, reference to which might serve in
lieu of the argument by the ubiquitous double reductio ad absurdum. 85
It was largely in connection with the search for some means of
simplifying the arguments in the tedious methods of the ancients that
the differential and integral calculus was developed in the seventeenth
century. To trace this development is the purpose of this essay; but
at this point it may not be amiss to anticipate the final formulation
to the extent of comparing the nature of the basic concept of the
calculus — that of the limit of an infinite sequence — with the view
indicated in the method of exhaustion.
The limit of the infinite sequence P h P 2 , . . . , P w , . . . (the
terms of which represent, for example, the areas of the inscribed
polygons considered in the proposition above) has been defined in the
introduction to be the number C, such that, given any positive num-
ber e, we can find a positive integer N, such that f or n > N it can
be shown that \ C — P n \ < e. The spatial intuition of the method
of exhaustion, with its application of areas, unlimited subdivision,
and argumentation by a reductio ad absurdum, here gives way to
definition in terms of formal logic and number, i. e., of infinite ordered
aggregates of the positive integers. The method of exhaustion corre-
sponds to an intuitional concept, described in terms of mental pic-
tures of the world of sensory perception. The notion of a limit, on
the other hand, may be regarded as a verbal concept, the explication
of which is given in terms of words and symbols — such as number,
infinite sequence, less than, greater than — with regard not to any
mental visualization, but only to their definition in terms of the
primary undefined elements. The limit concept is thus by no means
to be considered ineffable; nor does it imply that there is other than
84 Cf. Brunschvicg, Les tiapes de la philosophic mathtonatique, pp. 157-59.
86 The Works of Archimedes, ed. by T. L. Heath, p. cxliii.
Conceptions in Antiquity 37
empirical experience. It simply makes no appeal to intuition or
sensory perception. It resembles the method of exhaustion in that it
allows our vague instinctive feeling for continuity to shift for itself
in any effort that may be made to picture how the gap between the
curvilinear and the rectilinear, or between the rational and the irra-
tional, is bridged, for such an attempt is quite irrelevant to the logical
reasoning involved. The limit C is not for this reason to be regarded
as a sophistic or inconceivable quantity which somehow nevertheless
enters into real relations with other similar quantities, nor is it to
be visualized as the last term of the infinite sequence. It is to be
considered merely as a number possessing the property stated in the
definition. It is to be borne in mind that although adumbrations of the
limit idea appear in the history of mathematics in ancient times,
nevertheless the rigorous formulation of this concept does not appear
in work before the nineteenth century — and certainly not in the
Greek method of exhaustion. 86
The apparent break, in the mathematical work of Eudoxus, from
the metaphysical aspect of Platonism is seen equally clearly 87 in the
philosophical thought of one who studied under Plato for twenty
years and who was known as the "mind of the School." 88 Aristotle
borrowed freely from the work of his predecessors, with the result
that, although not primarily a mathematician, he was familiar with
the difficulties and results of Greek mathematics, including the method
of exhaustion. He wrote a work (now lost) On the Pythagoreans, dis-
cussed at some length the paradoxes of Zeno, mentioned Democritus
frequently in mathematics and science (although always to refute
him), was intimately familiar with Plato's thought, and was ac-
quainted with the work of Eudoxus. In spite of his competence in
mathematics and of his frequent use of geometry in his constructions, 89
Aristotle's approach to the problems involved was essentially scientific,
in the inductively descriptive sense. Furthermore, for Plato's mathe-
matical intellectualism he substituted a grammatical intuitionism. 90
86 Milhaud {Les Pkttosophes glomitres, p. 182) would have Eudoxus consider the circles
as the limit of a polygon, but this could not have been in the sense in which the term limit
is now employed.
87 Cf., in this respect, Jaeger, Aristoteles. w Ross, Aristotle, p. 2.
89 Enriques, Problems of Science, p. 110. See also Gorland, Aristoteles und die Mathe-
matik, and Heiberg, "Mathematisches zu Aristoteles."
90 Brunschvicg, Les Stapes de la philosophic mathimatique, p. 70.
38 Conceptions in Antiquity
Although he realized that the objects of mathematics are not those
of sense experience and that the figures used in demonstrations are for
illustration only 91 and in no way enter into the reasoning, Aristotle's
whole attitude was governed by a strong dependence upon the evi-
dence of the senses, as well as upon logic, and by an aversion to
abstraction and extrapolation beyond the powers of sensory percep-
tion. As a consequence, he did not think of a geometrical line, as had
Plato, as an idea which is prior to, and independent of, experience of
the concrete. Neither did he regard it, as does modern mathematics,
as an abstraction which is suggested, perhaps, though not in any
way defined, by physical objects. He viewed it rather as a character-
istic of natural objects which has merely been separated from its
irrelevant context in the world of nature. "Geometry investigates
physical lines but not qua physical," he said, 92 and added:
Necessity in mathematics is in a way similar to necessity in things which
come to be through the operation of nature. Since a straight line is what
it is, it is necessary that the angles of a triangle should equal two right
angles. 93
The decisions which Aristotle rendered on the indivisible, the infinite,
and the continuous were consequently those dictated by common
sense. In fact, with the exception of Plato's successors in the Academy
and, perhaps, of Archimedes, they were those accepted by the body of
Greek mathematicians after Eudoxus.
Only in the case of the indivisible, however, do Aristotle's views
coincide with the present notions in mathematics. Modern science has
opposed, modern mathematics upheld, Aristotle in his vigorous denial
of the indivisible, physical and mathematical, of the atomic school.
Recent physical and chemical theories of the atom have furnished a
description of natural phenomena which offers a higher degree of
consistency within itself and with sensory impressions than had the
Peripatetic doctrine of continuous substantiality. Science has conse-
quently, under Carneades' doctrine of truth, accepted the atom as a
physical reality. Modern mathematics, on the other hand, agrees with
Aristotle in his opposition to minimal indivisible line segments; not,
however, because of any argument from experience, but because it has
91 T. L. Heath, History of Greek Mathematics, I, 337.
« Physica II. 193b-194a. » Physica TL. 200a.
Conceptions in Antiquity 39
been unable to give a satisfactory definition and logical elaboration of
the concept. However, the mathematical indivisible, in spite of the
opposition of Aristotle's authority, was destined to play an important
part in the development of the calculus, which in the end definitely
excluded it. That the concept enjoyed an extensive popularity even in
Aristotle's day, Greek logic notwithstanding, is seen by the fact that
a Peripatetic treatise formerly ascribed to Aristotle (but now thought
to have been written by Theophrastus, or Strato of Lampsacus, or
perhaps by someone else), the De lineis insecabilibus, u was directed
against it. This presents many arguments against the assumption of
indivisible lines and concludes that "it conflicts with practically every-
thing in mathematics." 95
The work may have been composed as an answer to Xenocrates,
the successor of Plato in the Academy, who apparently maintained
the existence of mathematical indivisibles. It has been asserted 96 that
neither Aristotle nor Plato's successors understood the infinitesimal
concept of their master, and that only Archimedes rose to a correct
appreciation of it. It is to be remarked, however, that such an assump-
tion is wholly gratuitous. Plato in his extant works offered no clear
definition of the infinitesimal. Archimedes, moreover, made no men-
tion of any indebtedness to Plato in this matter, and, as will be-seen,
explicitly disclaimed any intention of regarding infinitesimal methods
as constituting valid mathematical demonstrations. 97 The opposition
of Aristotle to the doctrine of infinitesimals was wholly justified by
considerations of logic, although from the point of view of the subse-
quent development of the calculus the uncritical use of the infinitely
small was for a time most fruitful.
As in the case of the infinitesimal, so also with respect to the infinite
the views of Aristotle constitute excellent illustrations of his abiding
confidence in the ultimate interpretability of phenomena in terms of
distinctly clear concepts derived from sensory experience. 98 The
Pythagoreans had regarded space as infinitely divided, and Democritus
had likewise spoken of the atoms as infinite in number. Plato, influ-
M The Works of Aristotle, Vol. VI, Opuscula. K De lineis insecabilibtts 970a.
94 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 152.
91 The Method of Archimedes, ed. by T. L. Heath, p. 17.
w Aristotle's view of the infinite has been much discussed. For one of the most recent
philosophical discussions of this subject, see Edel, Aristotle's Theory of the Infinite.
40 Conceptions in Antiquity
enced by these views and perhaps also by those of Anaximander and
Anaxagoras, had not clearly distinguished the concrete from the
abstract." He had held that the infinite was located at the same time
in ideas and in the sensible world, 100 the line as made up of points being
one illustration of this fact. However, no one of Aristotle's predecessors
had made quite clear his position with respect to the infinite. Anaxag-
oras, at least, seems to have realized that it is only the imagination
which objects to the infinite and to an infinite subdivision. 101
On the other hand, Aristotle, in adopting the inductive scientific
attitude, did not go beyond what is clearly representable in the mind.
In consequence he denied altogether the existence of the actual infinite
and restricted the use of the term to indicate a potentiality only. 102
His clear distinction between an existent infinity and a potential
infinity was the basis of much of the discussion of the Scholastics on
the subject and of later controversies on the metaphysics of the
calculus. His refusal to recognize the actual infinite was in keeping
with his fundamental tenet that the unknowable exists only as a
potentiality: that anything beyond the power of comprehension is
beyond the realm of reality. Such a methodological definition of
existence has led investigators in inductive science to continue to the
present time the Aristotelian attitude of negation toward the infi-
nite; 103 but such a view, if adopted in mathematics, would exclude
the concepts of the derivative and the integral as extrapolations
beyond the thinkable, and would, in fact, reduce mathematical
thought to the intuitively reasonable.
That the Aristotelian doctrine of the infinite was abandoned in the
mathematics of the nineteenth century was largely the result of a
shift of emphasis from the infinite of geometry to that of arithmetic;
for in the latter field assumptions appear to be less frequently dictated
by experience. For Aristotle such a change of view would have been
impossible, inasmuch as his conception of number was that of the
Pythagoreans: a collection of units. 104 Zero was not included, of
course; nor was "the generator of numbers," the integer one. "The
99 Paul Tannery, Pour Vhistoire de la science Hellene, pp. 300-5.
100 Brunschvicg, Les Etapes de la philosophic mathimatique, p. 67.
101 Paul Tannery, Pour Vhistoire de la science Hellene, pp. 293-94.
102 Physica III. 206b. ** Barry, The Scientific Habit of Thought, p. 197.
104 Physica HI. 207b; cf. also Plato, Republic VII. 525e.
Conceptions in Antiquity 41
smallest number in the strict sense of the word 'number' is two,"
said Aristotle. 106 Such a view of number could not be reconciled with
the infinite divisibility of continuous magnitude which Aristotle
upheld so vigorously. When, then, Aristotle distinguished two kinds of
(potential) infinite — one in the direction of successive addition, or the
infinitely large, and the other in the direction of successive subdivision,
or the infinitely small — we find the behavior of number to be quite
different from that of magnitude:
Every assigned magnitude is surpassed in the direction of smallness, while
in the other direction there is no infinite magnitude .... Number on the
other hand is a plurality of "ones" and a certain quantity of them. Hence
number must stop at the indivisible. . . . But in the direction of largeness
it is always possible to think of a larger number. . . . Hence this infinite is
potential, .... and not a permanent actuality but consists in a process
of coming to be, like time .... With magnitudes the contrary holds.
What is continuous is divided ad infinitum, but there is no infinite in the
direction of increase. For the size which it can potentially be, it can also
actually be. 106
In commenting on the view of mathematicians, Aristotle said,
In point of fact they do not need the infinite and do not use it. They pos-
tulate only that the finite straight line may be produced as far as they
wish. . . . Hence, for the purposes of proof, it will make no difference to
them to have such an infinite instead, while its existence will be in the
sphere of real magnitude. 107
How well this characterizes Greek geometry can be seen in the method
of exhaustion as presented by Eudoxus slightly earlier than Aristotle
and by Euclid a little later. This method assumes in the proof only
that the bisection can be continued as far as one may wish, not car-
ried out to infinity. How far it lies from the point of view of modern
analysis is indicated by the fact that the latter has been called "the
symphony of the infinite."
That Aristotle would be unable to cope with the problems of the
continuous is perhaps to be expected, both from his view of the
infinite and from the lack among the Greeks of an adequate arith-
metical point of view. After having considered place and time in the
fourth book of the Physica, and change in the fifth, Aristotle turned
in the sixth book to the continuous. His account is based upon a
105 Physica IV. 220a. m Physica in. 207b. 1W Ibid.
42 Conceptions in Antiquity
definition derived from the intuitive notion of the essence of con-
tinuous magnitude: "By continuous I mean that which is divisible
into divisibles that are infinitely divisible." 108 This view was sup-
plemented by a naive appeal to the instinctive feeling of the necessity
of a hang-togetherness — of the coincidence of the extremities of the
component parts. 109 For this reason Aristotle denied that number can
produce a continuum, 110 inasmuch as there is no contact in numbers. 111
Only a generation ago this Aristotelian view had not been entirely
abandoned, 112 but the mathematical continuum accepted at the present
time is defined precisely in terms of the concept of number, or of
classes of elements, and of the notion of separation — as in the Dede-
kind Cut — rather than contact. The nature of continuous magnitude
has been found to lie deeper than Aristotle believed, and it has been
explained on the basis of concepts which require a broader definition
of number than that held during the Greek period. The Aristotelian
dictums on the subject were not unfruitful, however, for they led to
speculations during the medieval period which in turn aided in the
rise of the calculus and the modern doctrine of the continuum.
In connection with the study of continuous magnitude, Aristotle
attempted also to clarify the nature of motion, criticizing the atomists
for their neglect of the whence and the how of movement. 113 Although
he was quite adept at detecting problems, he failed to make his
formulation of these quantitative and was consequently infelicitous in
their resolution. In the light of modern scientific method, this lack of
mathematical expression gives to his treatment of motion and varia-
bility the appearance of a dialectical exercise, rather than of a serious
effort to establish a sound basis for the science of dynamics. 114
Aristotle's approach to the subject was qualitative and metaphys-
ical. This is evidenced by his definition of motion as "the fulfillment
of what exists potentially, in so far as it exists potentially," and by
the further remark, "We can define motion as the fulfillment of the
movable qua movable." 115 We shall find this qualitative explanation
of motion — the result of the striving of a body to become actually
what it is potentially — involved in less teleological forms in the idea
108 Physica VI. 232b. 1M Physica VI. 231a; cf. also Categories 5a.
110 Metaphysica 1075b and 1020a; Categoriae 4b. m Metaphysica 1085a.
112 See Mere, A History of European Thought in the Nineteenth Century, II, 644.
m Metaphysica 985b and 1071b.
114 Cf. Mach, The Science of Mechanics, p. 511. m Physica III, 201a-202a.
Conceptions in Antiquity 43
of impetus developed by the Scholastics, in Hobbes' explanation of
velocity and acceleration in terms of a conatus, and even in meta-
physical interpretations of the infinitesimal of the calculus as an
intensive quantity — that is, as a "becoming" rather than a "being."
In this respect Aristotle's work may have encouraged the elaboration
of notions leading toward the derivative. However, his influence was
in another sense quite adverse to the development of this concept in
that it centered attention upon the qualitative description of the change
itself, rather than upon a quantitative interpretation of the vague
instinctive feeling of a continuous state of change invoked by Zeno.
The calculus has shown that the concept of continuous change is no
more free from that of the discrete than is the numerical continuum,
and that it is logically to be based upon the latter, as is also the idea
of geometrical magnitude. As long as Aristotle and the Greeks con-
sidered motion continuous and number discontinuous, a rigorous
mathematical analysis and a satisfactory science of dynamics were
difficult of achievement.
The treatment of the infinite and of continuous magnitude found
in the Physica of Aristotle has been regarded as presenting the ap-
pearance of a veritable introduction to a treatise on the differential
calculus. 116 Such a view, however, is seen to be most unwarranted,
inasmuch as Aristotle expressed his unqualified opposition to the funda-
mental idea of the calculus — that of an instantaneous rate of change.
He asserted that "Nothing can be in motion in a present. . . . Nor
can anything be at rest in a present." 117 This point of view neces-
sarily operated against the mathematical representation of the phe-
nomena of change and against the development of the calculus.
Aristotle's denial of instantaneous velocity, as realized in the world
described by science, is, to be sure, in conformity with the recognized
limitations of sensory perception. Only average velocities, — , are
At
recognizable in this sense. In the world of thought, on the other hand,
it has been found possible — through the calculus and the limit con-
cept — to give a rigorous quantitative definition of instantaneous
. ds
velocity, — . Aristotle, however, in conformity with a view widely ac-
CLt
»• Moritz Cantor, "Origines du calcul infinit&imal," p. 6. u7 Physica VL 234a.
44 Conceptions in Antiquity
cepted at the time, regarded mathematics as a pattern of the world
known through the senses and consequently did not foresee such a
possibility.
The failure of Aristotle to distinguish sharply between the worlds of
experience and of mathematical thought resulted in his lack of clear
recognition of a similar confusion in the paradoxes of Zeno. Aristotle
refuted the arguments in the stade and the arrow by an appeal to
sensory perception and the denial of an instantaneous velocity.
Modern mathematics, on the other hand, has answered them in terms
of thought alone, based on the concept of the derivative. In the same
manner Aristotle resolved the paradoxes in the dichotomy and the
Achilles by the curt assertion, suggested by experience, that although
one cannot traverse an infinite space in finite time, it is possible to
cover an infinitely divided space in finite time because of the infinite
divisibility of the latter. 118
Mathematics has, of course, given the solution of the difficulties in
terms of the abstract concept of converging infinite series. In a certain
metaphysical sense this notion of convergence does not answer Zeno's
argument, in that it does not tell how one is to picture an infinite
number of magnitudes as together making up only a finite magnitude;
that is, it does not give an intuitively clear and satisfying picture, in
terms of sense experience, of the relation subsisting between the_
infinite series and the limit of this series. If one demands that Zeno's
paradoxes be answered in terms of our vague instinctive feeling for
continuity — as essentially different from the discrete — no answers
more satisfying than those of Aristotle (to whom we owe also the
statement of the paradoxes, since we do not have Zeno's words)
have been given. The unambiguous demonstration that the difficulties
implied by the paradoxes are simply those of visualization and not
those of logic was to require more precise and adequate definitions
than any which Aristotle could furnish for such subtle notions as
those of continuity, the infinite, and instantaneous velocity. Such
definitions were to be given in the nineteenth century in terms of the
concepts of the calculus; and modern analysis has, upon the basis of
these, clearly dissented from the Aristotelian pronouncements in this
field. The views of Aristotle are not on this account to be regarded —
™ Physica VI. 239b-240a.
Conceptions in Antiquity 45
as is all too frequently and uncritically maintained 119 — as gross mis-
conceptions which for two thousand years retarded the advancement
of science and mathematics. They were, rather, matured judgments
on the subject which furnished a satisfactory working basis for later
investigations which were to result in the science of dynamics and in
the mathematical continuum. Nevertheless, there is apparent in the
work of Aristotle the cardinal weakness of Greek logic and geometry:
a naive realism which regarded thought as a true copy of the external
world. 120 This caused him to place too ingenuous a confidence in cer-
tain instinctive feelings with respect to continuous magnitude and to
seek, of all possible representations, that which presented the greatest
plausibility in the light of sensory experience, rather than that which
offered the widest consistency in thought.
It has been said 121 that the fifth book of Euclid's Elements and the
logic of Aristotle are the two most unobjectionable and unassailable
treatises ever written. The two men were roughly contemporaries:
Aristotle lived from 384 to 322 b. c. ; Euclid's birth has been placed at
about 365 b. c. and the composition of the Elements may be accepted
as between 330 and 320 b. c. 122 There is also a marked similarity
between the Aristotelian apodictic and the mathematical method
built up by Euclid. 123 Although Euclid was probably taught by the
pupils of Plato, 124 the influence of the sober, hard-headed, scientific
thought illustrated by Eudoxus and Aristotle must have predominated
over the more abstract, speculative, and even mystical trend seen in
the immediate successors of Plato to the leadership of the Academy
and carried to excess by later Neoplatonists. There is in Euclid none
of the metamathematics which played such a prominent part in
Plato's thought, nor do metaphysical speculations on mathematical
atomism enter. Mathematics was regarded by Euclid neither as a
necessary form of cosmological intelligibility, nor as a mere tool of
119 See, for example, Mayer, "Why the Social Sciences Lag behind the Physical and Bio-
logical Sciences"; cf. also Toeplitz, "Das Verhaltnis von Mathematik und Ideenlehre bei
Plato."
120 Enriques, The Historic Development of Logic, p. 25.
121 By Augustus De Morgan. See Hill, "Presidential Address on the Theory of Pro-
portion."
122 See Vogt, "Die Lebenzeit Euklids."
123 Brunschvicg, Les £tapes de la philosophie mathematique, pp. 84-85.
124 T. L. Heath, History of Greek Mathematics, I, 356.
46 Conceptions in Antiquity
pragmatic utilitarianism. For him it had entered the domain of logic,
and in this connection Proclus tells us that Euclid subjected to rigor-
ous proofs what had been negligently demonstrated by his prede-
cessors. 126 Nevertheless, the Elements retained the realism which was
so clearly apparent in Aristotelian logic. 126
Although Aristotle had rejected Plato's doctrine of ideas, he had
retained a belief in a natural order of science and in the necessary
character of principles. This latter confidence was adopted likewise by
Euclid. Greek geometry was not formal logic, made up of hypothetical
propositions, as mathematics largely is today; but it was an idealized
picture of the world of actuality. Just as Aristotle seems not to have
clearly recognized the tentative character of scientific knowledge (thus
leaving himself open to the attacks of the Skeptics), so also he failed
to appreciate that although the conclusions drawn by mathematics
are necessary inferences from the premises, nevertheless the latter are
quite arbitrarily selected, subject only to an inner compatibility.
Aristotle considered hypotheses and postulates as statements which
are assumed without proof, but which are nevertheless capable of
demonstration. 127 Although he admitted that "we must get to know
the primary premises by induction" (rather than by pure intellection,
as Plato had believed), he maintained that "since except intuition
nothing can be truer than scientific knowledge, it will be intuition
that apprehends the primary premises," and primary premises are
therefore "more knowable than demonstrations." 128
The Euclidean view was similar to the Peripatetic attitude in giving
to geometry the characteristic form of logical conclusions from neces-
sary postulates. As such, it excluded any notions the nature of which
was not clearly and compellingly "felt" through intuition. The infinite
was never invoked in the demonstrations, true to Aristotle's statement
that it was unnecessary, its place being taken by the method of
exhaustion which had been developed by Eudoxus. The limitation of
the concept of number to that of positive integers apparently was
115 Cf . Proclus Diadochus, In primum Euclides elementorum librum commentariorum . . .
ibri IIII, p. 43.
m Enriques, The Historic Development of Logic, p. 25; cf. also Burtt, Metaphysical Foun-
dations of Modem Physical Science, p. 31.
m Analytica posterior a I. 76b.
m Analytica posterior a II. 100b; cf. also Brunschvicg, Les fttapes de la philosophic
mathlmatique, pp. 86-93.
Conceptions in Antiquity 47
continued, a broader view being made unnecessary by the Eudoxian
theory of proportion. 129
For Euclid ratio was not a number in the abstract arithmetical
sense (and in fact it did not become so until the time of Newton), 130
and the treatment of the irrational in the Elements is completely geo-
metrical. Furthermore, the axioms, postulates, and definitions of
Euclid are those suggested by common sense, and his geometry never
loses contact with spatial intuition. 131 His premises are the dictates
of sensory experience, much as Aristotle's science may be charac-
terized as a glorification of common sense. Such purely formal, logical
concepts as those of the infinitesimal and of instantaneous velocity, of
infinite aggregates and the mathematical continuum, are not elab-
orated in either Euclidean geometry or in Aristotelian physics, for
common sense has no immediate need for them. The ideas which were
to lead to the calculus had not in Euclid's time reached a stage' at
which a logical basis could have been afforded; mathematics had not
attained the degree of abstraction demanded for symbolic logic.
Although the origin of the notions of the derivative and the integral
are undoubtedly to be found in our confused thought about varia-
bility and multiplicity, the rigorous formulation of the concepts in-
volved, as we shall find, demanded an arithmetical abstraction which
Euclid was far from possessing. Even Newton and Leibniz, the inven-
tors of the algorithmic calculus, did not fully recognize the need for
it. The logical foundations of the calculus are much further removed
from the vague suggestions of experience — much more subtle — than
those of Euclidean geometry. Since, therefore, the ideas of variability,
continuity, and infinity could not be rigorously established, Euclid
omitted them from his geometry. The Elements are based on "refined
intuition," 132 and do not allow free scope to the "naive intuition"
which was to be especially active in the genesis of the calculus in the
seventeenth century. 133
From the point of view of the development of the calculus, there-
m See Stolz, Vorlesungen fiber allgemeine Arithmetik, I, 94; cf. also Schubert, "Principes
fondamentaux de l'arithmetique," pp. 8-9.
130 See Newton, Opera omnia, I, 2.
131 Cf. Barry, The Scientific Habit of Thought, pp. 215-17.
m As Felix Klein (The Evanston Colloquium Lectures on Mathematics, pp. 41-42) so aptly
puts it.
m Ibid.
48 Conceptions in Antiquity
fore, the Elements of Euclid show an uninteresting inflexibility of
rigor, discouraging to the growth of such new speculations and dis-
coveries. The work of Euclid represents the final synthetic form of all
mathematical thought — the elaboration by deductive reasoning of the
logical implications of a set of premises. Back of his geometry, how-
ever, stood several centuries of analytical investigation, carried out
often on the basis of empirical research, or on uncritical intuition, or,
not infrequently, on transcendental speculation. It was to be largely
from indagation of a similar type, rather than from the rigorously
precise thought of Euclid, that the development of the concepts of
the calculus was to proceed. This in its turn was necessarily to give
way, in the nineteenth century, to a formulation as eminently deduc-
tive — albeit arithmetic rather than geometric — as that found in the
Elements.
The greatest mathematician of antiquity, Archimedes of Syracuse,
displayed two natures, for he tempered the strong transcendental
imagination of Plato with the meticulously correct procedure of
Euclid. He "gave birth to the calculus of the infinite conceived and
brought to perfection successively by Kepler, Cavalieri, Fermat,
Leibniz, and Newton," 134 and so made the concepts of the derivative
and the integral possible. In the demonstration of his results, how-
ever, he adhered to the clearly visualized details of the Eudoxian
procedure, modifying the method of exhaustion by considering not
only the inscribed figure but the circumscribed figure as well. The
deductive method of exhaustion was not a tool well adapted to the
discovery of new results, but Archimedes combined it with infini-
tesimal considerations toward which Democritus and the Platonic
school had groped. The freedom with which he handled these is
shown most clearly in the treatise to which we have already referred,
the Method.™
This work, addressed to Eratosthenes the geographer, astronomer,
and mathematician of Alexandria, was lost and remained largely un-
134 Chasles, Aperqu historique sur Vorigine et le developpement des methodes en ge'ome'trie,
p. 22.
135 For the works of Archimedes in general, see Heiberg, Archimedis opera omnia and
T. L. Heath, The Works of Archimedes. For Archimedes' Method, see T. L. Heath, The
Method of Archimedes, Recently Discovered by Heiberg; Heiberg and Zeuthen, "Eine neue
Schrift des Archimedes"; and Smith, "A Newly Discovered Treatise of Archimedes."
Conceptions in Antiquity
49
known until rediscovered in 1906. In it Archimedes disclosed the
method which is presumably that which he employed in reaching
many of his conclusions in problems involving areas and volumes.
Realizing that it is advantageous to have a preliminary notion of the
result before carrying through a deductive geometrical demonstration,
Archimedes employed for this purpose, in conjunction with his law of
the lever, the idea of a surface as made up of lines. For example, he
showed that the truth of the proposition that a parabolic segment is
£ the triangle having the same base and vertex (the vertex of the
segment being taken as the point from which the perpendicular to the
base is greatest) is indicated by the following considerations from
FIGURE 2
mechanics. 136 In the diagram given in Figure 2, in which V is the
vertex of the parabola, BC is tangent at B, BD = DP, and X is any
point on AB, we know from the properties of the parabola that for
•*• *Y U .U .- XX '" AB BD DP ^
any position of X we have the ratio = = = . But
XX' AX DX" DX"
X" is the center of gravity of XX'", so that from the law of the lever
we see that XX', if brought to P as its midpoint, will balance XX'" in
its present position. This will be true for all positions of X on AB.
Inasmuch as the triangle ABC consists of the straight lines XX'" in
this triangle, and since the parabolic segment A VB is likewise made
"• See T. L. Heath, The Method of Archimedes, Proposition I, pp. 15-18.
50 Conceptions in Antiquity
up of the lines XX', we can conclude that the triangle ABC in its
present position will be in equilibrium at D with the parabolic seg-
ment when this is transferred to P as its center of gravity. But the
center of gravity of ABC is on BD and is \ the distance from D to B,
so that the segment A VB is \ the triangle A BC, or i the triangle A VB.
This method of Archimedes indicates an anticipation of the use of
the concept of the indivisible which was to be made in the fourteenth
century and which, when developed again more freely in the seven-
teenth century, was to lead directly to the procedures of the calculus.
The basis of the method is to be found in the assumption of Archimedes
that surfaces may be regarded as consisting of lines. We do not know
in precisely what sense he intended this to be understood, for he did
not speak of the number of elements in each figure as infinite, but
said rather that the figure is made up of all the elements in it. That he
probably thought of them as mathematical atoms is indicated not
only by this manner of expression, but also by the highly suggestive
fact that he was led to many new results by a process of balancing, in
thought, elements of dissimilar figures, using the principle of the
lever precisely as one would in weighing mechanically a collection of
thin laminae or material strips.
Using this heuristic method, Archimedes was able to anticipate the
integral calculus in achieving a number of remarkable results. He
discovered, among other things, the volumes of segments of conoids
and cylindrical wedges and the centers of gravity of the semicircle, of
parabolic segments, and of segments of a sphere and a paraboloid. 137
However, to assert that here "for the first time one can correctly speak
of an integration" 138 is to misinterpret the mathematical process
known by this name. The definite integral is defined in mathematics
as the limit of an infinite sequence and not as the sum of an infinite
number of points, lines, or surfaces. 139 Infinitesimal considerations,
similar to those in the Method, were at a later period to furnish per-
haps the strongest incentive to the development of the calculus, but,
m See The Works of Archimedes, Chap. VII, "Anticipations by Archimedes of the Inte-
gral Calculus"; also Method.
m Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 154; cf. also p. 155.
w T. L. Heath (The Method of Archimedes, pp. 8-9) correctly points out that the method
here used is not integration; but he gratuitously imputes to Archimedes the concept of the
differential of area.
Conceptions in Antiquity 51
as Archimedes realized, they lacked in his time all basis in rigorous
thought. This they continued to do until the concepts of variability
and limit had been carefully analyzed. For this reason Archimedes
considered that this method merely indicated, but did not prove,
that the result is correct. 140
Archimedes employed his heuristic method, therefore, simply as an
investigation preliminary to the rigorous demonstration by the method
of exhaustion. It was not a generous gesture that led Archimedes to
supplement his "mechanical method" by a proof of the results in the
rigorous manner of the method of exhaustion; it was, rather, a mathe-
matical necessity. It has been asserted that Archimedes' method
"would be quite rigorous enough for us today, although it did not
satisfy Archimedes himself." 141 Such an assertion is strictly correct
only if we ascribe to him our modern doctrines on number, limit, and
continuity. This ascription is hardly warranted, inasmuch as Greek
geometry was concerned with form rather than with variation. It was,
as a result, necessarily unable to frame a satisfactory definition of the
infinitesimal, which of necessity was to be regarded as a fixed quantity
rather than as an auxiliary variable. Archimedes was probably well
aware of the lack of any sound basis for his method and for this reason
recast all of his analysis by infinitesimals in the orthodox synthetic
form, much as Newton was to do almost nineteen hundred years later
after the methods of the calculus had been discovered but still lacked
adequate foundation.
The suggestive analysis of the problem of determining the area of
a parabolic segment had been given by Archimedes in the Method.
However, formal proofs (both mechanical and geometrical) of the
proposition were carried out by the method of exhaustion in another
treatise, the Quadrature of the Parabola.™ In these proofs Archimedes
followed his illustrious predecessors in omitting all reference to the
infinite and the infinitesimal. In the geometrical demonstration, for
example, he inscribed within the parabolic segment a triangle of area
A, having the same base and vertex as the segment. Then within each
140 T. L. Heath, History of Greek Mathematics, II, 29.
141 T. L. Heath, The Method of Archimedes, p. 10.
142 See The Works of Archimedes, and T. L. Heath, History of Greek Mathematics, II, 85-
91. A good adaptation of the geometrical proof is given in Smith, History of Mathematics,
II, 680-83.
52 Conceptions in Antiquity
of the two smaller segments having the sides of the triangle as bases,
he similarly inscribed triangles. Continuing this process, he obtained a
series of polygons with an ever-greater number of sides, as illustrated
(fig. 3). He then demonstrated that the area of the nth such polygon
was given by the series A (l + 4 + ^2 + . . . + 4»-i), where A
is the area of the inscribed triangle having the same vertex and base
as the segment. The sum to infinity of this series is %A, and it was prob-
ably from this fact that Archimedes inferred that the area of the para-
bolic segment was also §A , 143
However, he did not state the argument in this manner. Instead of
finding the limit of the infinite series, he found the sum of n terms and
added the remainder, using the equality
A i 1 + I + ? + ••• + I«-i + \ • i»-i) = \ A -
FIGURE 3
As the number of terms becomes greater, the series thus "exhausts"
$A only in the Greek sense that the remainder, j(^n-ijA, can be
made as small as desired. This is, of course, exactly the method of
proof for the existence of a limit, 144 but Archimedes did not so inter-
pret the argument. He did not express the idea that there is no re-
mainder in the limit, or that the infinite series is rigorously equal
to iA. m Instead, he proved, by the double reductio ad absurdum of
the method of exhaustion, that the area of the parabolic segment
could be neither greater nor less than iA . In order to be able to define
$A as the sum of the infinite series, it would have been necessary to
develop the general concept of real number. Greek mathematicians
did not possess this, so that for them there was always a gap between
the real (finite) and the ideal (infinite).
143 T. L. Heath, "Greek Geometry with Special Reference to Infinitesimals."
144 As Miller pointed out in "Some Fundamental Discoveries in Mathematics."
146 The Works of Archimedes, p. cxliii.
Conceptions in Antiquity 53
It is not strictly correct, therefore, to speak of Archimedes' geo-
metrical procedure as a passage to the limit, for the essential part of
the definition of a limit is the infinite sequence. 146 Inasmuch as he did
not invoke the limit concept, it is hardly correct to say that in finding
the sum of such series Archimedes answered in a very explicit and
definite manner some of the difficult questions raised by Zeno, and
that "These difficulties were completely solved by the Greek mathe-
maticians, and further serious arguments along this line seem to be
based upon ignorance or perversity." 147 The notion of the limit of an
infinite series is essential for the clarification of the paradoxes; but
Greek mathematicians (including Archimedes) excluded the infinite
from their reasoning. The reasons for this ban are obvious: intuition
could at the time afford no clear picture of it, and it had as yet no
logical basis. The latter difficulty having been removed in the nine-
teenth century and the former being now considered irrelevant, the
concept of infinity has been admitted freely into mathematics. The
related limit concept is now invoked in the explication of the para-
doxes, as well as in a simplification of Archimedes' long indirect
demonstrations.
The series given above is not the only one found in Archimedes'
work. In determining, by the method of exhaustion, the volume of a
segment of a paraboloid of revolution, he was led to investigations of
a similar nature. A consideration, in some detail, of the application
of the method which Archimedes here made 148 may be desirable at
this point, in order to bring out clearly the general character of his
procedure and to point out to what a remarkable extent it resembles
that used in the integral calculus, even though Archimedes did not
explicitly employ the limit concept.
Archimedes first circumscribed about the solid ABC (which he
called a conoid) the cylinder ABEF (fig. 4) having the same axis,
CD, as has the paraboloidal segment. He then divided the axis into
n equal parts of length h and through the points of division passed
planes parallel to the base. On the sections of the paraboloid thus
formed he constructed inscribed and circumscribed cylinder frusta, as
"• C. R. Wallner, "Ober die Entstehung des Grenzbegriffes," p. 250. See also Hankel,
"Grenze"; and Wieleitner, Die Geburt der Modemen Mathematik, II, 12.
147 Miller, "Some Fundamental Discoveries," p. 498.
148 On Conoids and Spheroids, Propositions 21, 22; The Works of Archimedes, pp. 131-33.
54
Conceptions in Antiquity
shown in the figure. He was then able to establish the equivalent of
the proportions:
Cylinder ABEF nVt
Inscribed figure h -\- 2h + . . . + (n — \)h
Cylinder ABEF n%
and —
Circumscribed figure h + 2h + . . . + nh
Archimedes had previously shown 149 (using a method cast in geo-
metrical form but otherwise much like that ordinarily employed in
elementary algebra to determine the sum of an arithmetic progression)
that h + 2h + ... + («- \)h < %n 2 h and that h + Ik + . . . +
nh > \rfih. At this point modern mathematics would employ the limit
C p
D
FIGURE 4
to become infinite,
= 2. Not so
concept and, allowing the series h + 2h -f 3h +
i • / ffth
would conclude that um (
»-»- \h + 2k + 3h + . . . + nh,
Archimedes. Instead of doing this, he showed that the proportions
above may be written, as a result of these inequalities,
Cylinder ABEF
Cylinder ABEF ^ 2 ,
> - and
1
2
< -.
1
inscribed figure 1 circumscribed figure
Now by the principle of exhaustion and the usual reductio ad absur-
dum, he concluded that the paraboloidal segment can be neither
greater than, nor less than, half the cylinder ABEF.
In the argument of Archimedes, the logical validity of a conclusion
149 In Proposition 10 of the work On Spirals. See Works of Archimedes, pp. 163-65.
Conceptions in Antiquity 55
based on the numerical concept of the limit of an infinite series was
not admitted, but was replaced by one based on the rigorous geomet-
rical method of exhaustion. It has been said that the differences in
the methods of infinitesimals, of exhaustion, and of limits are felt to
be more in the words than in the ideas; 150 but such an assertion may
lead to serious misinterpretation. The methods are, of course, inter-
related, and in consequence they lead to identical results; but the
points of view are distinctly different, as we have seen. Although in a
broad sense the procedures of Archimedes may be considered as
"practically integrations," 151 or even as representing in a general way
"an integration process," 152 it is indeed far from correct to speak of
.any one of them as a "veritable integration" 153 or as "the equivalent
of genuine integration." 154 The definite integral requires for its correct
formulation an appreciation of the notions of variability and func-
n
tionality, the formation of the characteristic sum 2 f(x { ) Ax i} and the
»=i
application of the concept of the limit of the infinite sequence obtained
from this sum by allowing n to increase indefinitely as Ax { becomes
indefinitely small. These essential aspects of the integral are, of
course, at no place indicated in the work of Archimedes, for they were
extrinsic to the whole of Greek mathematical thought.
The series h -f- 2h + 3h + . . . , employed by Archimedes in the
proposition above, was to figure prominently in the gropings toward
the calculus in the seventeenth century, so that it may be well to
point out that this geometrical demonstration is in broad outline
equivalent to performing the integration indicated by j^xdx. In
determining that the area bounded by the polar axis and by one
turn of the spiral P = — is £ that of the circle of radius a, 155 Archimedes
had occasion to make a similar calculation, equivalent to evaluating
j%x?dx. This was not done directly and arithmetically by determining
lim A 2 + ( 2h ) 2 + • • • + (nh) 2 \ ^ ,
lim ( L^_^! — ! — L_^ — L to be £, as he might easily
150 See Milhaud, Nouvelles etudes, p. 149.
161 T. L. Heath, History of Greek Mathematics, II, 3.
m Karpinski, "Is There Progress in Mathematical Discovery?" p. 48.
163 Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter, p. 181.
154 The Works of Archimedes, p. cliii; cf. also p. cxliii.
m On Spirals, Proposition 24, The Works of Archimedes, pp. 178-82.
56 Conceptions in Antiquity
have done had the Greeks not interdicted the infinite, but indirectly
and geometrically by coupling the inequalities
ff + (2hY + . . . + (nhY 1 W- + (2/Q 2 + ...+[(»- pffi
n(nh) 2 n(nh) 2
with the proof by the method of exhaustion, in a manner similar to
that employed in the proposition on the paraboloidal segment.
Archimedes may have known the equivalent result for the sum of
cubes as well, and much later the Arabs extended his work to include
also the fourth powers. In the seventeenth century at least a half dozen
mathematicians — Cavalieri, Torricelli, Roberval, Fermat, Pascal, and
Wallis — were to extend (all more or less independently) this work of
Archimedes still further by the determination of ffcfdx for yet other
values of n, and in this way to point immediately to the algorithm of
the calculus. The methods which these men were to use were not, in
general, the careful geometric procedures found in the propositions of
Archimedes, but were to be based on ideas of indivisibles and of
infinite series — notions which the decline of the idea of mathematical
rigor during the intervening years was to make more acceptable to
mathematicians, even though they were to be at that time no less
impeachable than in Archimedes' day.
We have seen that Greek geometry was concerned largely with
form rather than with variation, so that the function concept was
not developed. That motion was nevertheless occasionally invoked in
mathematics is indicated by Plato's suggestion that a line is gener-
ated by a moving point, as well as by the fact that certain special
curves, described by a double motion, had been discussed even before
Plato's time. The most famous of such curves was perhaps the quad-
ratrix of Hippias, the Sophist. Archimedes may have been influenced
by Hippias' idea when he denned his spiral as the locus of a point
which moved with uniform radial velocity along a line, while the line
in turn revolved uniformly about one of its end points which is kept
fixed. 156
In the further study of this curve Archimedes was led, in attempting
to determine its tangent, to one of the few considerations correspond-
ing to the differential calculus to be found in Greek geometry. In
conformity with the static geometry of design, a tangent to a circle
166 The Works of Archimedes, p. 165.
Conceptions in Antiquity 57
had been denned by Euclid 167 as a line touching the circle at only
one point, and this definition was extended by Greek geometers to
apply to other curves as well. There is also ascribed to the ancients 158
the definition, following the suggestion contained in a proposition of
Euclid, 189 of a tangent as a line touching a curve and such that in
the space between the straight line and the curve no other straight
line can be interposed. These definitions were, of course, of restricted
applicability and did not, in general, suggest a method of procedure
for drawing tangents. Although Archimedes did not offer a more satis-
factory definition, he appears, nevertheless, to have employed, for the
determination of the tangent to his spiral, a method suggestive of a
more general point of view. As in quadratures he had used considera-
tions from the science of statics, so here in the problem of tangents
Archimedes appears to have had recourse to a representation derived
from kinematics. It seems likely, although Archimedes did not thus
express the idea, that he found the tangent to the spiral p = ad
through a determination of the instantaneous direction of motion of
the point P, by which it is traced. 160 This he probably did by applying
to the motions by which the spiral may be generated the parallelogram
of velocities, the principle of which had been perceived by the Peri-
patetics. 161 The motion of P may be regarded as compounded of two
resultant motions: one with a radial velocity of constant magnitude
V r directed along the line OP (see fig. 5), and the other in a direction
perpendicular to this and having a magnitude which is given by the
variable product, V a , of the distance OP and the uniform speed of
rotation. Inasmuch as the distance OP and the speeds are given, the
parallelogram (in this case a rectangle) of velocities can be con-
structed and the direction of the resultant velocity, and therefore also
the tangent PT, determined.
The determination by Archimedes of the tangent to the spiral has
been characterized as "a differentiation," 162 or as "corresponding to
" Book III, Definition 2, in T. L. Heath ed., II, 1.
168 See, for example, Comte, The Philosophy of Mathematics, pp. 108-10.
"» Book III, Proposition 16, in T. L. Heath ed., II, 37.
180 T. L. Heath, History of Greek Mathematics, II, 556-61.
U1 See Mechanica XXXIII. 854b-855a in The Works of Aristotle, Ross ed., Vol. VI,
Opuscula; cf. also Duhem, Les Origines de la statique, II, 245; and Mach, The Science of
Mechanics, p. 511.
1M Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 116.
58
Conceptions in Antiquity
our use of the differential calculus." 163 Such a designation, however,
is hardly justified. Apparently, Archimedes made no effort to develop
the idea here involved into a uniform method of attack upon the
problem of tangents to other curves. When, in the seventeenth cen-
tury, the method again appeared in the work of Torricelli, Roberval,
Descartes, and Barrow, the scope of its applicability was somewhat
extended; but only with the method of fluxions of Newton was there
presented an algorithmic procedure for determining from the equa-
tion of any curve a pair of generating motions from which the tangent
might be found. There was in Greek geometry no idea of a curve as
FIGURE 5
corresponding to a function, nor was there a satisfactory definition
of a tangent in terms of the limit concept. There was therefore in the
thought of Archimedes no anticipation of the realization that the
geometrical notion of tangency is to be based upon the function con-
cept and upon the numerical idea of a limit, i. e., upon the expression
lim / /(* + *) "/(*) ) wh i ch f urri ishes the basis for the differ-
ential calculus. There is in the whole of Greek mathematics no clear
recognition of the need for the limit concept, both for the determina-
1M T. L. Heath, History of Greek Mathematics, II, 557.
Conceptions in Antiquity 59
tion of curvilinear areas and of tangents to curves, and even for the
very definition of these ideas which intuition vaguely suggests.
It is therefore incorrect to impute to Archimedes the ideas ex-
pressed in the integral and the derivative. These notions were not
part of Greek geometry. Nevertheless, the problems and methods of
Archimedes furnished probably the strongest incentive to the later
development of such ideas, for they were leading in precisely this
direction. The work of Archimedes so strongly suggests the newer
methods of analysis that in the seventeenth century Torricelli and
Wallis hazarded the opinion that the ancient Greek mathematicians
had deliberately concealed under their synthetic demonstrations the
analytic devices by which they had been led to their discoveries. 164
Through the discovery of the Method of Archimedes, the assumption
of the existence of such methods has been proved correct, but the
failure of ancient geometers further to elaborate these in their works
is not to be regarded as indicating an intent to deceive. Although
the notions of instantaneous velocity and of the infinitely small were
accepted — all too uncritically — by Torricelli, Wallis, and their con-
temporaries, these ideas were not considered by Greek thinkers as
admissible in mathematics. In the seventeenth century, however, the
infinitesimal and kinematic methods of Archimedes were made the
basis of the differential and the fluxionary forms of the calculus.
Although Leibniz and Newton were thus to admit into the calculus
the notions of the infinitely small and of instantaneous velocity,
these were to remain open to criticism even after that — in fact, until,
in the nineteenth century, the basic concept, that of the derivative,
had been carefully defined.
After the time of Archimedes, the trend of Greek geometry was
toward applications, rather than toward new theoretical develop-
ments. 165 No Greek mathematician approached nearer to the calculus
than had Archimedes. His successors — Hipparchus, Heron, Ptolemy,
and others — turned to various mathematical sciences, such as as-
tronomy, mechanics, and optics. Nevertheless, the infinitesimal con-
siderations of Archimedes were not forgotten by the Greek geometers
of later times. Toward the end of the third century of our era, in a
"< T. L. Heath, History of Greek Mathematics, II, 21, 557; Torricelli, Opere, I (Part 1), 140*
" 5 T. L. Heath, History of Greek Mathematics, II, 198.
60 Conceptions in Antiquity
period which is regarded as one of decline in mathematics, the geom-
eter Pappus not only displayed a familiarity with these methods,
but was able also to add a new result — known as the Pappus theorem —
to the work of Archimedes on centers of gravity. 166
However, further significant advances in geometrical method were
to be dependent upon certain broad changes in other branches of
mathematics. One of these was the development of a more highly
elaborated abstract symbolic algebra; another was the introduction
into algebra and geometry of the notion of variation — of variables
and functionality. The Arithmetic of Diophantus, which represents the
highest development of Greek algebraical thought, was really theo-
retical logistic, 167 rather than generalized arithmetic or the study of
certain functions of variables. In it the Peripatetic basis in logic is
stronger than the Platonic ontological conception of mathematics.
Only one unknown was introduced in this work, and the only solu-
tions which were accepted as having meaning were those which, in ac-
cordance with Aristotelian tradition, could be expressed as the quotient
of integers. Irrational and, imaginary numbers were not recognized. 168
The Arithmetic of Diophantus represents a "synocopated algebra,"
in that abbreviations for certain recurring quantities and operations
are systematically introduced in it. However, in order that his work
might be associated with geometrical results and later serve as the
suitable basis for the calculus, it had to be made more completely
symbolic, the concept of number had to be generalized, and the ideas
of variable and function had to be introduced. During the Middle
Ages the interest of the Hindus and the Arabs in algebraic develop-
ment, and the attack by the Scholastic philosophers upon the prob-
lems offered by the continuum, were to supply in some measure the
background required for these changes. When, therefore, in the
sixteenth and seventeenth centuries, the classical work of Archimedes
was developed into the methods constituting the calculus, the advance
was made along lines suggested by the traditions built up during the
medieval period.
184 We cannot be sure, however, that the theorem was an original contribution on the
part of Pappus or that he possessed a proof of it. See Pappus of Alexandria, La Collection
tnathematique, trans, by Ver Eecke; cf. also Weaver, "Pappus."
167 Jakob Klein, "Die griechische Logistik und die Entstehung der Algebra"; see also
Tannery, Pour la science hellene, p. 405.
188 T. L. Heath, History of Greek Mathematics, I, 462.
III. Medieval Contributions
THE MOOT points as to the origin and antiquity of Hindu
mathematics are not immediately pertinent in a precis of the
development of the derivative and the integral; for these concepts
depend on certain logical subtleties, the significance of which appears
to have surpassed the appreciation, or at least to have escaped the
interest, of the early Indian mathematicians.
The Hindus apparently were attracted by the arithmetical and com-
putational aspects of mathematics, 1 rather than by the geometrical
and rational features of the subject which had appealed so strongly
to the Hellenic mind. Their name for mathematics, ganita, meaning
literally the "science of calculation," 2 well characterizes this prefer-
ence. They delighted more in the tricks that could be played with
numbers than in the thoughts the mind could produce, so that neither
Euclidean geometry nor Aristotelian logic made a strong impression
upon them. The Pythagorean problem of the incommensurable, which
was of intense interest to Greek geometers, was of little import to
Hindu mathematicians, who treated rational and irrational quantities,
curvilinear and rectilinear magnitudes indiscriminately. 3 With respect
to the development of algebra, this attitude occasioned perhaps an
incidental advance, since by the Hindus the irrational roots of quad-
ratics were no longer disregarded, as they had been by the Greeks,
and since to the Hindus we owe also the immensely convenient con-
cept of the absolute negative. 4 These generalizations of the number
system and the consequent freedom of arithmetic from geometrical
representation were to be essential in the development of the con-
cepts of the calculus, but the Hindus could hardly have appreciated
the theoretical significance of the change.
Similarly, another consequence of the lack of nice distinction in
Hindu thought happens to have been responsible for a change which
corresponds to the modern view. We have seen that the crisis of
1 Cf. Karpinski, The History of Arithmetic, p. 46.
2 Datta and Singh, History of Hindu Mathematics, Part I, p. 7.
3 Lasswitz, Geschichte der Atomistik, I, 185.
4 Fine, The Number System of Algebra, p. 105.
62 Medieval Contributions
the incommensurable led to the abandonment by the Greeks of the
attempt to associate numbers with all geometrical magnitudes. The
Pythagorean problem of the application of areas was in general insolu-
ble in terms of the conceptions of number and geometrical magnitude
then current. The area of the circle could not literally be exhausted
by applying rectilinear configurations to it, inasmuch as curvilinear
magnitudes were fundamentally different. The area of a circle was
not to be compared with that of a square, for area was not a numerical
concept, and equality in general — as distinct from equivalence —
meant congruence. The number * would have had no meaning in
Greek mathematics. With the Hindus the view was different. They
saw no essential unlikeness between rectilinear and curvilinear figures,
for each could be measured in terms of numbers; arithmetic and
mensuration, rather than geometry and considerations of congruence,
were fundamental. The strong Greek distinction between the discrete-
ness of number and the continuity of geometrical magnitude was not
recognized, for it was superfluous to men who were not bothered by
the paradoxes of Zeno or his dialectic. Questions concerning incom-
mensurability, the infinitesimal, infinity, the process of exhaustion,
and other inquiries leading toward the conceptions and methods of
the calculus were neglected.
Operational difficulties were felt in dealing with the number zero —
difficulties which led Brahmagupta to regard zero as an infinitesimal
quantity which ultimately reduces to nought; and which caused
Bhaskara to say that the product of a number and zero is zero, but
that the number must be retained as a multiple of zero if any further
operations impend. 5 Yet these difficulties do not appear to have been
considered with the intention of resolving the logical questions, im-
plicit in the use of indeterminate forms, which were later to puzzle
the early users of the calculus. Questions of limits, although implied
in their work, were not expressly stated. 6 The emphasis which Hindu
mathematicians placed on the numerical aspect of the subject, to-
gether with the use of the Hindu numerals and of the principle of
positional notation (the latter having been employed also by the
Babylonians) did, of course, make more easily possible the develop.
5 Datta and Singh, op. cit., p. 242.
6 Sengupta, "History of the Infinitesimal Calculus in Ancient and Mediaeval India,"
p. 224.
Medieval Contributions 63
ment of algebra, and, subsequently, that of the algorithmic procedures
of the calculus. However, the logical concepts of the derivative and
the integral are just as easily denned in terms of Greek numeration
as of our own, so that Hindu mathematics added no thought essential
to the development of these ideas.
The Hindu numerals reached Europe through the medium of the
Arabic civilization. This was preeminently eclectic, so that in Arabian
mathematics we find both Greek and Hindu elements. The Arabic
reckoning is based on that of Hindus, and Arabic trigonometry is Indian
also in its use of the sine and the arithmetic form, rather than of the
Hipparchan chord and geometric representation. Arabic geometry,
however, shows the influence of Euclid and Archimedes, and Arabic
algebra indicates a return to Greek geometric demonstration and the
Diophantine avoidance of negative numbers. 7 The general character
of Arabic algebra, however, is somewhat different from that of Pio-
phantus' Arithmetic, for it is rhetorical rather than syncopated and
deals mostly with problems of practical life, rather than with the
abstract properties of numbers. 8 The whole trend of Arabic mathe-
matics was, like that of the Hindus, directed away from the specula-
tions on the incommensurable, continuity, the indivisible, and infin-
ity — ideas which in Greek geometry were leading toward the calculus.
Furthermore, additions to the classic Greek works — such as the
treatise we have by Alhazen (or Ibn al-Haitham) on the measurement
by infinitesimals of the paraboloid and on the summations of the
cubes and the fourth powers of the positive integers 9 — were slight,
for Arabic thought lacked the interest which was necessary to pursue
further such fecund ideas. To the Arabs, however, we owe the preserva-
tion and transmission to Europe of much of the Greek work which
would otherwise have been lost.
Christian Europe had, since the time of Pappus, added practically
nothing to traditional mathematical theory, and was, in fact, largely
unfamiliar with the ancient treatises until, in the twelfth century,
Latin translations were made from Arabic, Hebrew, and Greek
7 Fine, Number System; cf. also Paul Tannery, Notions historiques, p. 333, and Karpinski,
Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi, p. 21.
8 Gandz, "The Sources of al-Khowarizmi's Algebra," pp. 263-77.
9 See Ibn al-Haitham (Suter), "Die Abhandlung iiber die Ausmessung des Paraboloids."
Cf. also Wieleitner, "Das Fortleben der archimedischen Infinitesmalmethoden bis zum
Beginn der 17. Jahr., insbesondere ueber Schwerpunkt bestimmungen."
64 Medieval Contributions
manuscripts. Before this time the work of Euclid was known chiefly-
through the enunciation, largely without proofs, of selected proposi-
tions given in the early sixth century by Boethius, in his Geometry. 10
The work of Archimedes fared no better, for by the sixth century the
only works of his generally known (and commented upon by Eutocius)
were those on the sphere and cylinder, on the measurement of a circle,
and on the equilibrium of planes (that is, on the law of the lever). 11
When in the twelfth and thirteenth centuries the Greek works
began to appear in Latin translations, they did not meet with an
enthusiastic reception on the part of European scholars, interested as
these men were in theology and metaphysics. The interest of Roger
Bacon and the appearance of works such as those of Jordanus
Nemorarius show that in the thirteenth century there was no apathet-
ical lack of mathematical activity; but the knowledge displayed at
this time indicates an inadequate familiarity with the classic works of
Greek geometry 12 — a nescience which had led to the designation fuga
miserorum 13 — for the fifth proposition of Euclid's Elements. 14,
The inadequate attention paid to Greek geometry during the later
medieval period was paralleled by a similar lack of zeal for Greek
and Arabic algebraic methods. The thirteenth century opened aus-
piciously with the appearance, in 1202, of the Liber abaci of Leonardo
of Pisa. This, however, was not followed by a comparable work for
almost 300 years — that is, until the appearance in 1494 of the Summa
de ariihnietica of Luca Pacioli. The dearth of advances in the mathe-
matical tradition during this period has been made the basis for
very severe strictures on the mathematical work done in this inter-
val. 15 Such condemnation is justified by thinking in terms of contri-
10 Ball, A Short Account of the History of Mathematics, p. 107.
11 See The Works of Archimedes, p. xxxv.
12 Ginsburg, "Duhem and Jordanus Nemorarius," p. 361.
13 Later pons asinorum.
M Smith, "The Place of Roger Bacon in the History of Mathematics," pp. 162-67.
15 Hankel (Zur Geschichte der Mathematik im Alterthum mid Mittelalter, p. 349) says:
"Mit Erstaunen nimmt man wahr, dass das Pfund, welches einst Leonardo der latein.
ischen Welt iibergeben, in diesen drei Jahrhunderten durchaus keine Zinsen getragen
hatte; wir finden, von Kleinigkeiten abgesehen, keinen Gedanken, keine Methode, welche
nicht aus dem liber abaci oder der practica geometriae bereits wohl bekannt oder ohne
Weiteres abzuleiten ware." See also pp. 357-58 for a similar complaint. Cajori (A History
of Mathematics, p. 125) says that in this period "the only noticeable advance is a simplifi-
cation of numerical operations and a more extended application of them." Similarly,
Archibald (Outline of the History of Mathematics, p. 27) characterizes the interval from
Medieval Contributions 65
butions to Greek geometry 16 and to Arabic algebra (to which, however,
Nicole Oresme did add in the fourteenth century the conception of a
fractional power 17 which was later to add to the facility of application
of the methods of the calculus). During the interval from 1202 to
1494 there may, indeed, have appeared no successor to either Archi-
medes or to Leonardo of Pisa. 18 If, on the other hand, we regard the
broader aspects of mathematics — the speculations and investigations
which lead up to the propositions which are in the end deductively
demonstrated — it will appear that this so-called barren period fur-
nished points of view of significance in the development of the cal-
culus. In this respect, as in others, there was perhaps as much origi-
nality in medieval times as there is now. 19
Throughout the early period of the Middle Ages, Aristotle had
been known in Europe largely through his logical works. During the
thirteenth century, however, his scientific treatises circulated freely
and, although these were condemned at Paris in 1210, 20 their study
was again established in the university by 1255, at which time nearly
all of Aristotle was prescribed for candidates for the master's degree. 21
In the Physica, Aristotle had considered in some detail the infinite,
the infinitesimal, continuity, and other topics related to mathe-
matical analysis. These became, particularly in the next century, the
center of a lively discussion on the part of scholastic philosophers.
They were studied in the light of Peripatetic philosophy, rather than
in terms of mathematical postulational thought, but the resulting
speculations were of service in sustaining an interest in such con-
ceptions until, at a later date, they became a part of mathematics.
Leonardo of Pisa to Regiomontanus as "a period of about 250 barren years." Bjornbo
("Uber ein bibliographisches Repertorium der handschriftlichen mathematischen Literatur
des Mittelalters," p. 326) says "Von einer Entwickelung in dieser Epoche ist kaum zu
reden; bedeutende mathematische Fortschritte wird man hier vergebens suchen."
14 It has been well said by Sarton {Introduction to the History of Science, I, 19-20) that
we do not do justice to the medievals because we judge their first steps by the Greek last
steps.
17 Cf. Algorhmus proportionum, pp. 9-10. Fine {Number System, p. 113) goes so far as
to say that this is the only contribution of the period to algebra.
18 Enestrom, "Zwei mathematische Schulen im christlichen Mittelalter."
19 See Sarton, op. cit., I, 16.
^Denifle and Chatelain, Ckartularium Universitatis Parisiensis, Vol. I, p. 70, No. 11.
Cf. also Vol. I, pp. 78-79, No. 20.
21 Ibid., Vol. I, pp. 277-79, No. 246. Cf. also Rashdall, The Universities of Europe in the
Middle Ages, I, 357-58.
66 Medieval Contributions
One of the best examples of Scholastic thought on these topics is
that found in the work of a man who exerted a great influence upon
medieval thought, 22 Thomas Bradwardine, "doctor profundus" and
Archbishop of Canterbury, and perhaps the greatest English mathe-
matician of the fourteenth century. In his Geometria speculativaP and
in the Tractatus de continuo 2i Bradwardine discussed, among other
things, the nature of continuous magnitude, his view being dominated
by the Peripatetic opposition to any atomism. The doctrine of Leucip-
pus and Democritus, which had denied divisibility to infinity, has at
all times had partisans and adversaries, and the Scholastic period
was far from exceptional in this respect. The idea of the indivisible,
during the earlier Middle Ages, was often more elementary than that
held by Democritus long before. It seems to have been believed by
Capella, Isidore of Seville, Bede, and others, that time is composed
of indivisibles, an hour being made up of 22,560 such instants. 25 It
seems probable, or at least possible, that these instants were regarded
as atoms of time. During the later medieval period the idea of indi-
visibles under various forms and modifications was upheld by Robert
Grosseteste, Walter Burley, and Henry Goethals, among others. 26 On
the other hand, Roger Bacon protested, in his Opus majus, that the
doctrine of indivisibles was inconsistent with that of incommensura-
bility, 27 an argument developed further by Duns Scotus, William of
Occam, Albert of Saxony, Gregory of Rimini, and others. 28 Brad-
wardine considered, in the light of the problem of the continuum, the
divers points of view represented by proponents of the doctrine of
indivisibles. Some interpreted the question in terms of physical
atomism, others of mathematical points; some assumed a finite, others
an infinite, number of points; some postulated immediate contiguity,
others a discrete set of indivisibles. 29 Bradwardine himself maintained
22 Duhem, Les Origines de la statiqae, II, 323.
23 For brief indications of the contents of this work, see Moritz Cantor, Vorlesungen, II,
103-6; and Hoppe, "Zur Geschichte der Infinitesimalrechnung," pp. 158-60.
24 For an analysis of this see Stamm, "Tractatus de continuo von Thomas Bradwardina";
see also Cantor, Vorlesungen, II, 107-9.
25 Paul Tannery, "Sur la division du temps en instants au moyen age," p. 111.
24 Stamm, "Tractatus de continuo," pp. 16-17; cf. also Duhem, Etudes sur Leonard de
Vinci, II, 10-18.
27 Smith, The Place of Roger Bacon in the History of Mathematics, p. 180, n.
28 Duhem, Etudes sur Leonard de Vinci, II, 8.
29 Stamm, "Tractatus de continuo," p. 16.
Medieval Contributions 67
that continuous magnitudes, although including an infinite number
of indivisibles, are not made up of such atoms. 30 "Nullum continuum
ex indivisibilibus infinitis integrari vel componi," said Bradwardine, 31
using perhaps for the first time in this connection the word which
Leibniz was to adopt (upon the suggestion of the Bernoulli brothers 32 )
to designate in his calculus the sum of an infinite number of infinites-
imals — the integral. Bradwardine asserted, on the contrary, that a
continuous magnitude is composed of an infinite number of continua
of the same kind. The infinitesimal, therefore, evidently possessed for
him, as for Aristotle, only potential existence. 33
William of Occam seems to have occupied a position intermediate
between that of Bradwardine and that held by the supporters of
indivisible lines. While admitting that no part of any continuum is
indivisible, he maintained that, contrary to the teachings of Aristotle,
the straight line does actually (not only potentially) consist of points. 34
In another connection, however, Occam said that points, lines, and
surfaces are pure negations, having no reality in the sense that a
solid is real. 35 One must not, moreover, read too much into the Scho-
lastic views on the nature of the continuum. The opinion of Brad-
wardine has been rather freely identified with that of Brouwer and
the modern intuitionists, who conceive of the continuum as made up
of an infinite number of infinitely divided continua; 36 the view of
Occam has been said to correspond to that held by Russell and the
formalists, who regard the continuum as a perfect set of points every-
where dense. 37 Such comparisons are justifiable only in a very general
sense, for the Scholastic speculations invariably centered upon the
metaphysical question of the reality of indivisibles, rather than upon
the search for a representation which should be consistent with the
premises of mathematics. There was in the medieval views no con-
ception of the rigorous axiomatic foundation of arithmetic which has
30 Moritz Cantor, Vorlesungen, II, 108.
3I Moritz Cantor, Vorlesungen, II, 109, n.; Hoppe, "Zur Geschichte der Infinitesimal-
rechnung," p. 159; cf. also Stamm, op. cit., p. 17.
32 James Bernoulli, "Analysis problematis antehac propositi," p. 218.
M Lasswitz, Geschichte der Atomistik, II, 201; Stamm, "Tractatus de continuo," p. 17.
34 Burns, "William of Ockham on Continuity."
38 Duhem, £tudes sur Leonard de Vinci, II, 16-17; III, 26.
38 Stamm, "Tractatus de continuo," p. 20.
37 See Birch, "The Theory of Continuity of William of Ockham," p. 496.
68 Medieval Contributions
characterized modern thought upon this subject. Nevertheless, the
fourteenth century disputations on the indivisible represent a keen
appreciation of the difficulties involved and a clarity of thought which
was, several centuries later, to lend an air of respectability to the
infinitesimal methods leading to the calculus.
In discussions of indivisibles, the question of infinite division and
the nature of the infinite arose, of necessity. In fact, the medieval
philosophers discussed the question more from the point of view of
infinite divisibility and infinite aggregates than from that of infinitely
great magnitudes. Aristotle, it will be recalled, had distinguished two
kinds of infinity — the potential and the actual. The existence of the
latter he had categorically denied, and the former he had admitted
as realized only in cases of infinitely small continuous magnitudes and
of infinitely large numbers. 38 The Roman poet Lucretius had, with
keen imagination, upheld the notion of the infinite as indicating more
than the potentiality of indefinite increase. In focusing attention
upon infinite multitudes rather than magnitude, he adumbrated a
number of properties of infinite aggregates, such as that a part may
in this case be equal to the whole. 39 The work of Lucretius was not,
however, familiar to the European scholars of the Middle Ages. The
Aristotelian distinction, on the other hand, was continued by the
Scholastic philosophers, although with modifications resulting, per-
haps, from the fact that Christianity recognized an infinite God. In
the thirteenth century Petrus Hispanus, who became Pope John XXI,
recognized in his Summulae logicales** two kinds of infinity: a cate-
gorematic infinity, in which all terms are actually realized, and a
syncategorematic infinity, which is bound up always with potential-
ity. 41 This distinction is not greatly different from that which has
been suggested recently by a mathematician and scientist who would
discriminate between saying that an infinite aggregate is conceivable
and saying it is actually conceived. 42
The discussion of the two infinities, which was begun in the thir-
teenth century, was continued throughout the fourteenth also. Albert
nphysica, Book III.
" See Keyser, "The R6le of the Concept of Infinity in the Work of Lucretius."
40 Duhem, Etudes sur Uonard de Vinci, H, 22.
41 Duhem, op. cit., Vol. II, IAonard de Vinci et les deux infinis, pp. 1-53, gives an exten-
sive account of this question.
** Enriques, Problems of Science, pp. 127-28.
Medieval Contributions 69
of Saxony, for example, brought out the distinction nicely by a mere
transposition of words, saying that the two views were illustrated
respectively by the sentences: "in infinitum continuum est divisible,"
and "Continuum est divisible in infinitum." 43 Bradwardine brought
out the difference, perhaps with less subtlety but certainly more
clearly, in saying that the categorematic infinity is a quantity with-
out end, whereas the syncategorematic infinity is a quantity which is
not so great but that it can be made greater. 44
Although the distinction between the two infinities was generally
recognized by the philosophers of the Scholastic period, there were
significant differences, then as now, on the question of their existence.
William of Occam, in conformity with his nominalistic attitude and
with the principle of economy enunciated in his well-known "razor,"
agreed with Aristotle in denying that the categorematic infinity is
ever realized. Gregory of Rimini, on the other hand, maintained 45
what mathematics in the nineteenth century was to demonstrate:
that there is in thought no self-contradiction involved in the idea of
an actual infinity — the so-called completed infinite.
More interesting from the mathematical point of view than these
philosophical and discursive discussions are the remarks made on the
subject of the infinite by Richard Suiseth, popularly known as the
Calculator, in his Liber calculationum. This was composed later than
1328, inasmuch as it refers to Bradwardine's 46 Liber de proportionibus
of that year, 47 and probably dates from the second quarter of the
fourteenth century. 48 Although more interested in dialectical argu-
43 Duhem, Etudes sur Leonard de Vinci, II, 23.
44 Stamm, "Tractatus de continuo," pp. 19-20.
46 See Duhem, Etudes sur Leonard de Vinci, II, 399-401.
46 Duhem {Etudes sur Leonard de Vinci, III, 429) has incorrectly asserted that Calculator
names in his work only Bradwardine, Aristotle, and Averroes, but there are in the Liber
calculationum references also to ancient mathematicians, for example, Euclid (fol. 29 r ,
cols. 1-2) and Boethius (fol. 43", col. 1).
47 "Ut venerabilis magister Thomas de berduerdino in suo libro de proportionibus
liquide declarat." Liber calculationum, fol. 3", col. 1. Professor Lynn Thorndike has kindly
allowed me to make use of his rotograph of a copy in the British Museum, I B. 29, 968,
fol. l r — 83.* This is a copy of the undated editio princeps at Padua, placed by Thorndike
(A History of Magic and Experimental Science, III, 372) in the year 1477. Inasmuch as
this work is neither well known nor readily available, passages from it will be cited at
some length. The printed text contains so many abbreviations that it is difficult to read,
but transcriptions from it will be given in full.
48 See Thorndike, History of Magic, III, 375. Stamm ("Tractatus de continuo," p. 24)
would place its appearance after 1350.
70 Medieval Contributions
ments and subtle sophisms concerning infinity than in its adequate
definition, Suiseth made several comments on the subject which are
of particular mathematical significance. In the second chapter he said
that all sophisms regarding the infinite could be easily resolved by
recognizing that a finite part can have no ratio to an infinite whole. 49
This conclusion, he said, would be conceded by the imagination, for
the contrary would imply that any part, when added to the whole,
would not change it in magnitude. Arguments with respect to the
infinite do not proceed, therefore, as do those concerning finite
quantities. 50
About two hundred years later Galileo remarked still more clearly
the essential difference between the rules for the finite and those for
the infinite, but he focused attention upon the correspondence between
infinite aggregates, rather than upon the ratio of finite to infinite
magnitudes — a change of view which led to the final formulation of the
calculus in the nineteenth century. There is in the statements of
Calculator a cogency and a warning which might well have been
observed in later centuries, but they display the Peripatetic propensity
to regard the infinite as a magnitude, rather than as an aggregation
of terms. The fruitfulness of the infinite in the work of Archimedes
arose out of the infinite collections of lines or of terms in a series.
Reference will be made below, to be sure, to the study by Calculator
of an infinite series, but in this connection it will be seen that it was
not the endlessness of the sequence of terms which most interested
him, but a certain infinite magnitude. A consideration of this series
will require, however, the study of a larger problem with which it was
associated, and to which we shall now turn.
The blending of theological, philosophical, mathematical, and sci-
entific considerations which has so far been evident in Scholastic
thought is seen to even better advantage in a study of what was per-
49 "Infinite quasi sophismata possunt fieri de infinite que omnia si diligenter inspexeris
quod nullius partis ad totum infinitum est aliqua proportio faciliter dissoluere poteris per
predicta." Liber calculationum, fol. 8", col. 2.
60 "Que potest concedi de inmaginatione et causa est quia nulla pars finite finite intensa
respectu tocius infiniti aliquid confert quia nullam habet proportionem ad illud infini-
tum si tamen subiectum esset finitum conclusio non foret inmaginabilis quia tunc conclusio
inmediate repugnaret illi positioni quia tunc quelibet pars in comparatione ad totum
conferet aliquantum et sic non est nunc ideo nullum argumentum proceditur de infinite
sicut faceret de finite." Ibid., fol. 8 r , col. 2, fol. 8 s , col. 1.
Medieval Contributions 71
haps the most significant contribution of the fourteenth century to
the development of mathematical physics. It has commonly been
protested that additions, if any, which were made to scientific knowl-
edge in the medieval period lay solely in the field of practical dis-
coveries and applications; and that the only mathematical achieve-
ment during this time was the simplification of the rules of operation
for the Hindu-Arabic numerals, the latter having been made known
in Europe by Leonardo of Pisa and other men of the thirteenth cen-
tury. There is at least one exception to such an assertion, for it was
precisely during this interval, and particularly in the fourteenth cen-
tury, that a theoretical advance was made which was destined to be
remarkably fruitful in both science and mathematics, and to lead in
the end to the concept of the derivative. This consisted in the idea —
often expressed, to be sure, in terms of dialectical rather than mathe-
matical method — of studying change quantitatively, and thus ad-
mitting into mathematics the concept of variation. 51
Heraclitus, Democritus, and Aristotle had made some qualitative
metaphysical speculations on the subject of motion, and occasionally
Greek geometers (Hippias, Archimedes, Nicomedes, Diodes) had al-
lowed this notion to enter their thoughts (though not their proofs);
but the idea of representing continuous variation by means of geo-
metrical magnitude or of studying it in terms of the discreteness of
number does not seem to have arisen with them. The Greek sciences
of astronomy, optics, and statics had all been elaborated geometrically,
but there was no such representation of the phenomena of change.
Archimedes' famous work in statics was not paralleled by any equiv-
alent kinematic system which admitted of representation in the form
of mathematical propositions.
Perhaps the demand for rigor in Greek thought, which made con-
gruence fundamental in geometry and which allowed no confusion of
51 It is preposterous to assert — as does Tobias Dantzig (Aspects of Science, p. 45) —
that, with reference to Aristotle and the Schoolmen, "his casuistry, his predilection for the
static, his aversion for everything that moved, changed, flowed, or evolved admirably
suited their purposes." A brief examination of the works of Aristotle on physical science
and a glance at A Catalogue of Incipits of Mediaeval Scientific Writings in Latin by Thorn-
dike and Kibre will show how untrue is such a statement. Aristotle made extensive investi-
gations, from the point of view of physics, into the phenomena of motion. Scholastic
philosophers not only continued his work, but also added the quantitative form of ex-
pression which was so successfully developed later in the seventeenth century.
72 Medieval Contributions
the discrete with the continuous, could not be met by any attempt to
establish a science of dynamics. Moreover, Greek astronomy lacked
the concept of acceleration; the motions involved were all uniform
(and hence eternal) and circular, and could, in this case, be represented
by the geometry of the circle. No such uniformity was apparent for
local motion — that is, for terrestrial changes of position. Motion was,
it appeared, a quality rather than a quantity; and there was among
the ancients no systematic quantitative study of such qualities. 82
Aristotle had spoken of mathematics as concerned with "things
which do not involve motion," and had held that mathematics
studies objects qua continuous, physics qua moving, and philosophy qua
being. 63 In general Greek mathematics was the study of form, rather
than variability. The quantities entering into Diophantine algebraic
equations are constants, rather than variables, and this is true also
of Hindu and Arabic algebra. In the Scholastic period, however, there
arose a problem which was ultimately to change this view. Aristotle
had considered motion a quality that does not increase and decrease
through the joining together of parts, as does a quantity, 64 and this
idea dominated most thought until toward the end of the thirteenth
century, 66 at which time a reaction against Peripateticism arose, which
was to lead, at Paris, to new views on the subject of motion. 66
Bacon, in the second half of the thirteenth century, still followed
the Aristotelian discussion of motion, 67 but a new approach to the prob-
lem was evidenced early in the fourteenth century by the introduction
of the idea of impetus — the notion that a body, once set in motion,
will continue to move because of an internal tendency which it then
possesses, rather than, as Peripatetic doctrine had taught, because of
the application of some external force, such as that of the air, which
continues to impel it. This doctrine, which has been ascribed to Jean
Buridan, 68 was particularly significant as an adumbration of the
famous work in dynamics of Galileo almost three centuries later. At
52 Brunschvicg, Les Stapes de la philosophic mathe'matique, p. 97.
w Physica II. 198a; Metaphysica 1061.
84 Duhem, Eludes sur Ltonard de Vinci, III, 314-16; cf. also Wieleitner, "Ueber den
Funktionsbegriff und die grapbische Darstellung bei Oresme," pp. 196-97.
56 Wieleitner, "Ueber den Funktionsbegriff," p. 197.
'• Duhem, Etudes sur Leonard de Vinci, II, pp. iii-iv.
67 Thomson, "An Unnoticed Treatise of Roger Bacon on Time and Motion."
58 See Duhem, Etudes sur Ltonard de Vinci, passim.
Medieval Contributions 73
the time of its inception it served also to make more acceptable the
intuitive notion of instantaneous velocity, an idea excluded by
Aristotle from his science, but implied by the quantitative study of
variation of the fourteenth century. At that stage no precise definition
of instantaneous rate of change could, of course, be given — nor was
one given by Galileo — but there appeared at the time a large number
of works, more philosophical than mathematical, all based on the
intuition of this concept which everyone thinks he possesses. These
tractates were devoted to a discussion of the latitude of forms, that
is, of the variability of qualities.
There seems to be no scientific term which correctly expresses the
equivalent of the word form as here used. It refers in general to any
quality which admits of variation and which involves the intuitive
idea of intensity — that is, to such notions as velocity, acceleration,
density. These concepts are now expressed quantitatively in terms of
limits of ratios — that is, simply as numbers — so that no need is now
felt for a word to express the medieval idea of a form. In general, the
latitude of a form was the degree to which the latter possessed a cer-
tain quality, and the discussion centered about the intensio and the
remissio of the form, or the alterations by which this quality is ac-
quired or lost. Aristotle had distinguished between uniform and non-
uniform velocity, but the critical dialectical treatment of the Scho-
lastics went much further. In the first place, the time rates of change
which they considered were not necessarily those of distance, but
included many others as well, such as those of intensity of illumina-
tion, of thermal content, of density. Secondly — and more signifi-
cantly — they distinguished not only between latitudo uniformis and
latitudo difformis (that is, between uniform and nonuniform rates of
change), but proceeded further to classify the latter as either latitudo
uniformiter difformis or latitudo difformiter difformis (that is, according
as the instantaneous rate of change of the rate of change was uniform
or not); and the last-mentioned were sometimes in turn further
divided into either latitudo uniformiter difformiter difformis or latitudo
difformiter difformiter difformis.
These attempts to introduce order, by means of verbal arguments,
into the disconcerting problem of variability were destined to be
replaced centuries later by equivalent statements, expressed in mathe-
74 Medieval Contributions
matics and science by the remarkably concise terminology and sym-
bolism of algebra and the differential calculus. At the time at which
they were made, however, they represented the first serious and
careful effort to make quantitative the idea of variability.
The origin of the question of the latitude of forms is shrouded in
doubt. Duns Scotus appears to have been among the first to consider
the increase and the decrease (intensio and remissio) of forms, 69
although the loose idea of latitude of forms apparently goes back to
some time before this, inasmuch as Henry Goethals used the word
latitudo in this connection. 60 In the early part of the fourteenth cen-
tury there appeared works on variability and the latitude of forms by
James of Forli, Walter Burley, and Albert of Saxony. 61 Similar ideas
appeared also in 1328, at Oxford, in the treatise on proportions of
Bradwardine. 62 This work is devoted more particularly to mechanics
than to arithmetic, but the archbishop did not make a special investi-
gation into the theory of the latitude of forms. 63
"In the fourteenth century the study of the mathematical sciences
flourished greatly in Oxford," 64 and it was here that appeared not
only the work of Bradwardine, but also the "leading model" 66 of such
treatises on the latitude of forms — the Liber calculationum of Suiseth
to which reference has already been made. 66
That the doctrine of the latitude of forms was well known by the
middle of the fourteenth century is indicated by the fact that Cal-
culator begins in medias res by a general consideration, in the first
chapter, of the intension and remission of forms and the question as
to whether a latitudo difformis corresponds to its maximum or minimum
gradus or intensity. The sense in which a form could correspond to
any gradus is not clear, but Calculator appears here to be striving
59 Von Prantl, Geschichte der Logik im Abendlande, III, 222-23. Cf. Stamm, "Tractatus
de continuo," p. 23.
60 Duhem, Etudes sur Leonard de Vinci, III, 314-42.
61 Wieleitner, "Der 'Tractatus de latitudinibus formarum des Oresme,' " pp. 123-26.
82 Duhem, Etudes sur Leonard de Vinci, III, 290-301.
68 Stamm, "Tractatus de continuo," p. 24.
84 Gunther, Early Science in Oxford, II, 10.
86 Thorndike, History of Magic, III, 370.
88 For a general description of the contents of this work, see Thorndike, History of
Magic, Vol. Ill, Chap. XXIII; and Duhem, Etudes sur Lionard de Vinci, III, 477-81.
No analysis of the Liber calculationum from the mathematical point of view appears to be
available.
Medieval Contributions 75
toward the idea of average intensity, an idea which could not be
made precise without the use of the concepts of the calculus. In
Chapter II, however, he arrived, in this connection, at a result which
was to be of particular significance in the later development of science
and mathematics. Here, in considering the intension in difform things,
he reached the conclusion, in connection with problems on thermal
content, that the average intensity of a form whose rate of change
over an interval is constant, or of a form which is such that it is uni-
form throughout each half of the interval, is the mean of its first and
last intensities. 67 The rigorous proof of this requires the use of the
limit concept, but Calculator had resort to dialectical reasoning, based
on physical experience of rate of change. He argued, in connection
with such a form, that if the greater intensity is allowed to decrease
uniformly to the mean while the lesser is increased at the same rate
to this mean, then the whole is neither increased nor decreased. 68 The
argument is pursued at further length here, and the author adds, as
well, numerical illustrations such as the following: If the intensity
increases uniformly from four to eight, or if for the first half of the
time it is four, and for the last half it is eight, then the effect is that
which would result from a uniform intensity of six, operating through-
out the whole time. 69
The methods presented here are amplified and applied, in other
chapters of the Liber calculationum, to questions dealing with density,
velocity, and the intensity of illumination. In these Suiseth again
made use of the law that if the rate of change of one of these is con-
stant, or if the form is such that in each of the two halves of the
67 "Primo arguitur latitudinem caliditatis uniformiter difformem seu etiam difformem
cuius utraque medietas est uniformis, et calidum uniformiter difforme suo gradui medio
correspondere." Liber calculationum, fol. 4", col. 2.
68 "Capiatur talis caliditas seu tale calidum et remittatur una medietas ad medium et
intendatur alia ad medium equeuelociter: et sequitur totum non intendi nee remitti; eo
quod totam latitudinem acquiret. Secundum unam medietatem seu partem sicut deperdet
secundum aliam partem equalem et in fine erit uniforme sub tali gradu medio. Igitur nunc
correspondet tali gradui." Ibid.
69 "Sit enim tale uniformiter difforme seu difforme cuius utraque medietas est uniformis
una ut. VIII. et alia ut JUL gratia argumenti. Tunc prima qualitas ut .VIII. extenditur
per medietatem totius per predicta. Ergo solum denominat totum ut quatuor per idem
prima qualitas ut quatuor per aliam medietatem extensa solum facit ut duo ad totius
denominatorem. Igitur ille due qualitates totum precise denominabunt ut .VI. qui est
gradus medius inter illas medietates. Sequitur igitur positio sic in speciali." Ibid., fol. 5*,
col. 2.
76 Medieval Contributions
interval the rate of change is zero, then the average intensity is the
mean of the first and last values. As was the case in the study of
variation of the intensity of heat, no definition is given of the terms
employed, inasmuch as this would presuppose an appreciation of the
limit concept. The lack of such precise definitions led Calculator into
difficulties involving the infinite. He considered, for example, a rarity
of degree zero as a density of infinite degree, and conversely, 70 and
consequently became involved unnecessarily often in the paradoxes of
the infinite.
The tendency displayed by Calculator to consider the infinite from
the point of view of intensity or magnitude, rather than aggregation,
is brought out again in the discussion of an example of nonuniform
variation, which might have had a significant influence upon the
development of the calculus, had Suiseth's purpose been less that of
introducing mathematics into dialectical discussions of change and
more that of bringing the problem of variation into the realm of
mathematics. Calculator, in the second book of his Liber calculationum,
had occasion to consider the following problem: if throughout half
of a given time interval a variation continues at a certain intensity,
throughout the next quarter of the interval at double this intensity,
throughout the following eighth at triple this, and so ad infinitum;
then the average intensity for the whole interval will be the intensity
of the variation during the second subinterval (or double the initial
intensity). 71
This is equivalent to a summation of the infinite series % + f +
ft
I + tV + ...+ — +... = 2. It will be recalled that Archimedes
70 "Ergo .a. est infinite densum et per consequens non est rarum. Ex istis sequitur ista
conclusio quod aliquid est rarum quod non est rarum quia .a. est rarum quia est uniformiter
difforme rarum et non est rarum quia est infinite densum. Pro istis negatur utraque con-
clusio et tunc ad casum positum quod .a. sit unum uniformiter difformiter rarum terminatur
ad non gradum raritatis negatur casus nam ex quo raritas se habet privative sequitur ut
argutum est quod ab omni gradu raritatis usque ad non gradum raritatis est latitudo
infinita quia non gradus raritatis est infinitus gradus densitatis sed impossibile est quod
aliquid sit uniformiter difforme aliquale mediante latitudine infinita. Ideo casus est im-
possibilis sicut est impossibile quod aliquid sit uniformiter difforme remissum ad non
gradum remissionis terminatum." Ibid., fol. 19", cols. 1-2.
71 "Contra quam positionem et eius fundamentum arguitur sic quia sequitur quod si
prima pars proportionalis alicuius esset aliqualiter intensa, et secunda in duplo intensior,
et tertia in triplo intensior, et sic in infinitum totum esset equale intensum precise sicut est
secunda pars proportionalis quod tamen non uidetur visum." Ibid., fol. 5 , col. 2.
Medieval Contributions 77
had made use of certain simple series in connection with his geometry*
but that the characteristic Greek interdiction of the infinite had led
him to consider the sum to n terms only. It was apparent to him that
the series 1 + I + tz + ti + • ■ - + -p-i approached $ in such a
way that the difference could be made, by taking a sufficiently large
number of terms, less than any specified quantity. He did not, how-
ever, go so far as to define i as the "sum" of the infinite series, for this
would have exposed his thought to the paradoxes of Zeno, unless he
had invoked the precisely formulated concept of a limit as given in
the nineteenth century. The Scholastic discussions of the fourteenth
century, on the other hand, referred frequently to the infinite, both as
actuality and as potentiality, with the result that Suiseth, with per-
fect confidence, invoked an infinite subdivision of the time interval
to obtain the equivalent of an infinite series. He did not resolve the
aporias of Zeno, to show in what sense an infinite series may be said
to have a sum — a problem which future mathematicians were to con-
sider at length. Calculator, instead, was more particularly interested
in infinite magnitudes than in infinite series. Not only is the time
interval in his problem infinitely divided, but the intensity itself
becomes infinite. Now how can a quantity, whose rate of change
becomes infinite, have a finite average rate of change? Suiseth admitted
that this paradoxical result was in need of demonstration and so
furnished at great length the equivalent of a proof of the convergence
of the infinite series. This he did as follows.
Consider two uniform and equal rates of change, a and b, operating
throughout a given time interval, which has been subdivided in the
ratios \, \, \, . . . . Now let the rate of change b be doubled through-
out the interval; but in the case of a, let it be doubled in the second
subinterval; tripled in the third; and so to infinity, as given in the
problem above. Now the increase in a in the second subinterval, if
continued constantly throughout this and all following subintervals,
would result in an increase in the effect equal to that brought about
by the change in b during the first half of the time. The tripling of a
in the third subinterval, if continued constantly throughout this and
the ensuing subintervals, would in turn result in a further increase in
the effect of a equal to that brought about by the change in b in the
second subinterval, and so to infinity. Hence the increase resulting
78 Medieval Contributions
from the doubling, tripling, and so forth of a is equal to that caused
by the doubling of b; i. e., the average rate of change in the problem
considered above is the rate of change during the second subinterval,
which was to be proved. 72
The tediously verbose proof given by Calculator is, of course, based
entirely on arguments appealing to our intuition of uniform rate of
change. Because Suiseth gave no unambiguously clear definitions of
velocity, density, intensity of illumination, and other terms used
freely in his dissertation, his work not infrequently presents — as had
also, to a certain extent, the Physica of Aristotle 73 — the appearance of
an effort to propound sophistical questions on the subject of change,
rather than that of a serious effort to establish a scientific basis for
the study of the phenomena of motion and variability. 74
n "Nam apparet quod ilia qualitas est infinita ergo si sit sine contrario infinite denomina-
bit suum subiectum. Et quod conclusio sequatur arguitur sic: sint .a. .b. duo uniforma
eodem gradu, et dividatur .b. in partes proportionales et est ilia hora ita quod partes
maiores terminentur seu incipiant ab hoc instanti, et ponatur quod in prima parte pro-
portionali illius hore intendatur prima pars .b. ad duplum, et in secunda parte proportionali
intendatur secunda pars proportionalis illius ad duplum, et sic in infinitum, ita quod in fine
erit .b. uniforme sub gradu duplo ad gradum nunc habitum. Et ponatur quod .a. in prima
parte proportionali illius hore intendatur totum residuum .a. prima parte proportionali
.a. acquirendo totam latitudinem sicut tunc acquireret prima pars proportionalis .b. et in
secunda parte proportionali euisdem hore intendatur totum residuum .a. a prima parte
proportionali et secunda illius .a. acquirendo tantam latitudinem sicut tunc acquiret pars
proportionalis secunda .b. et in tertia parte proportionali indendatur residuum a prima
parte proportionali et secunda et tertia acquirendo tantam latitudinem sicut tunc acquiret
tertia pars proportionalis .b. et sic in infinitum scilicet quod quandocumque aliqua pars
proportionalis .b. intendetur pro tunc intendatur .a. secundum partes proportionales
subsequentes partem correspondentem in .a. acquirendo tantam latitudinem sicut acquiret
pars prima in .b. et sint .a.b. consimilia quantitatiue continue quo posito sequitur quod
.a. et .b. continue equeuelociter intendentur quia .a. continue per partes proportionales
similiter intendetur sicut .b. quia residuum a prima parte proportionali .a. est equale
prime parti proportionali eidem. Cum igitur .b. in prima parte proportionali illius hore
continue intendetur per primam partem proportionalem et .a. per totum residuum a prima
sua parte proportionali patet quod .a. in prima parte proportionali equeuelociter intendetur
cum .b. et sic de omni alia parte eo quod quandocumque .b. indendetur per aliquam
partem proportionalem .a. intendetur per totum interceptum inter partes correspondentes
sibi et extremum ubi partes terminantur scilicet minores. Cum ergo quelibet pars propor-
tionalis cuiuslibet continui sit equalis toti intercepto inter eandem et extremum ubi partes
minores terminantur. Igitur patet quod .a. continue equeuelociter intendetur cum .b. et
nunc est eque intensum cum .b. ut ponitur in casu. Ergo in fine .a. erit .a. eque intensum
cum .b. et .a. tunc est tale cuius prima pars proportionalis erit aliqualiter intensa, et
secunda pars proportionalis in duplo intensior, et tertia in triplo intensior et sic in infini-
tum. Et .b. erit uniforme gradu sub quo erit secunda pars proportionalis .a. ergo sequitur
conclusio." Ibid., fol. 5", col. 2— fol. 6 f , col. 1.
n Cf. Mach, The Science of Mechanics, p. 511.
m "Multa alia possent fieri sophismata per rarefactionem subiecti, et per fluxum quali-
tatis et alterations qualitatis si subiectum debet intendi et remitti per huiusmodi rare-
Medieval Contributions 79
Suiseth's unfortunate adoption of the Peripatetic attitude that such
qualities as dryness, coldness, and rarity are the opposites of moist-
ness, warmth, density — rather than simply degrees of the latter 75 —
complicated his consideration of these through the unnecessarily fre-
quent introduction of the paradoxes of the infinite. Nevertheless, to
Calculator we owe perhaps the first serious effort to make quanti-
tatively understandable these concepts of mathematical physics. His
bold study of the change of such quantities anticipated not only the
scientific elaboration of these, but also adumbrated the introduction
into mathematics of the notions of variable quantity and derivative.
In fact, the very words jluxus and jluens, which Calculator used in
this connection, 76 were to be employed by Newton some three hun-
dred years later, when in his calculus he spoke of such a variable
mathematical quantity as a fluent and called its rate of change a
fluxion. Newton apparently felt as little need as Suiseth for a defi-
nition of this notion of fluxion, and was satisfied to make a tacit appeal
to our intuitions of motion. Our definitions of uniform and nonuniform
rate of change, are, as Suiseth anticipated, numerically expressed; but
their rigorous definition could be given only after the development,
to which Newton contributed, of the limit concept. This latter arose
out of the notions of the calculus, which, in their turn, had evolved
from the intuitions of geometry. The prolix dialectic of Calculator made
no appeal to the geometrical intuition, which was to act as an inter-
mediary between his early attempts to study the problem of varia-
tion and the final formulation given by the calculus. This link between
the interminable discursiveness of Suiseth and the concise symbolism
of algebra was supplied by others of the fourteenth century who
studied the latitude of forms. Of these the most famous was
Oresme.
Nicole Oresme was born about 1323 and died in 1382. He was thus
somewhat younger than Calculator, from whom we have a manuscript
factionem, et fluxum et alterationem, ad que omnia considerando proportionem totius ad
partem responsionem alicere poteris ex predictis faciliter." Liber calculationum, fol. 9 r , col.
2. The word fluxum has been italicized by me for emphasis.
76 It should be remarked, in this connection, that several centuries later Galileo entered
into an extensive discussion as to whether the Peripatetic qualities are to be regarded as
positive (Le Opere di Galileo Galilei, I, 160 ff.), and that even in the eighteenth and nine-
teenth centuries the question as to the existence of "frigerific" rays was still argued. See,
for example, Neave, "J ose Ph Black's Lectures on the Elements of Chemistry," p. 374.
76 Liber calculationum, fol. 9 r , col. 2; fol. 75", col. 1.
80 Medieval Contributions
dated 1337, 77 and his doctrines were in all probability derived from
those of Suiseth and others of the Oxford School. 78 Furthermore, from
the very character of their work it would seem reasonable to place the
composition of the Liber calculationum at Oxford before the appear-
ance of the work of Oresme at Paris, on the grounds that the latter
lacks the chief defects of the former — the prolixity of the complicated
dialectical demonstrations and the author's propensity for subtle
sophisms. There is in the entire Liber calculationum no diagram or
reference to geometrical intuition, the reasoning being purely verbal
and arithmetical. 79 On the other hand, Oresme felt that the multi-
plicity of types of variation involved in the latitude of forms is dis-
cerned with difficulty, unless reference is made to geometrical figures. 80
The work of Oresme therefore makes most effective use of geometrical
diagrams and intuition, and of a coordinate system, to give his
demonstrations a convincing simplicity. This graphical representation
given by Oresme to the latitude of forms marked a step toward the
development of the calculus, for although the logical bases of modern an-
alysis have recently been divorced as far as possible from the intuitions
of geometry, it was the study of geometrical problems and the attempt
to express these in terms of number which suggested the derivative
and the integral and made the elaboration of these concepts possible.
The Tractatus de latitudinibus formarum* 1 has been generally
77 See Thorndike, History of Magic, III, 375.
78 Duhem (fLtudes sur Leonard de Vinci, III, 478-79) would see in the Liber calcula-
tionum the influence of Oresme and would characterize the work of Suiseth as "l'ceuvre
d'une science senile et qui commence a radoter." Wieleitner ("Zur Geschichte der unend-
lichen Reihen im christlichen Mittelalter," pp. 166-67) refers to Duhem in saying that
Calculator rediscovered some of Oresme's examples. Thorndike has clearly shown, how-
ever, that Oresme's work was subsequent to that of Calculator.
79 Stamm ("Tractatus de continuo," p. 23), apparently referring to an edition which
appeared in 1520 and which he mistakenly holds to be the first, asserts that Suiseth illus-
trated his representation with geometrical figures. Such illustrations may well have been
later interpolations, inasmuch as none such appear in the first edition (1477). Furthermore,
French copyists sometimes drew diagrams on the edge of manuscripts of such Oxford
works (Duhem, Etudes sur Leonard de Vinci, III, 449).
80 "Quia formarum latitudines multipliciter variantur que multiplicitas difficiliter dis-
cernitur nisi ad figuras geometricas consideratio referatur." Oresme, Tractatus de lati-
tudinibus formarum, fol. 201 r . I have made use of a photostat of a manuscript of this work
in the Bayerische Staatsbibliothek, Miinchen, cod. lat. 26889, fol. 201''— fol. 206'. See
also Funkhouser, "Historical Development of the Graphical Representation of Statistical
Data," p. 275; cf. also Stamm, op. cit., p. 24.
81 See Wieleitner, "Der 'Tractatus'," for a comprehensive description of this work. For
comments on the work of Oresme and a study of this period as a whole, see Duhem,
fctudes sur Leonard de Vinci, III, 346-405.
Medieval Contributions 81
ascribed to Oresme and cited as his chief work on the subject. Although
the views expressed in it are probably to be attributed to him, the
treatise appears to be only a poor imitation, probably by a student, 82
of a larger work by Oresme entitled Tractatus de figuratione potentiarum
et mensurarum.™ This latter work, one of the fullest on the latitude of
forms, was written probably before 1361. 84 It opens with the repre-
sentation of variation by means of geometry, rather than with a dia-
lectical exposition in terms of number, such as Calculator had given.
Following the Greek tradition, which regarded number as discrete and
geometrical magnitude as continuous, Oresme was led naturally to
associate continuous change with a geometrical diagram. An intension,
or rate of change by which a form acquires a quality, he imagined as
represented by a straight line drawn perpendicular to a second line,
the points of which represent the divisions of the time or space inter-
val involved. 85 For example, a horizontal line or longitude may rep-
resent the time or duration of a given velocity, and the vertical
height or latitude the intensity of the velocity. 86 It will be seen that
the terms "longitude" and "latitude" are here used to represent,
in a general sense, what we should now designate as the abscissa and
the ordinate. The work of Oresme does not, of course, represent the
earliest use of a coordinate system, for ancient Greek geography had
employed this freely; nor can his graphical representation be regarded
as equivalent to our analytic geometry, 87 for it lacks the fundamental
« Duhem, Etudes sur Uonard de Vinci, III, 399-400.
83 This work is known also by other titles, such as De uniformitale et difformitate inien-
tionum, and De conjlguratione qualitatum. See Wieleitner, "Ueber den Funktionsbegriff,"
for an extensive description of the portions of this work which are of significance in mathe-
matics. For an analysis of those sections of the De configuratione qualitatum which deal
with magic, see Thorndike, History of Magic, Vol. Ill, Chap. XXVI.
84 Wieleitner, "Ueber den Funktionsbegriff," p. 198; Duhem, Etudes sur Uonard de
Vinci, III, 375.
85 "Omnis res mensurabilis extra numeros ymaginatur ad modum quantitates continue.
Ideo opportet pro ejus mensuratione ymaginare puncta, superficies et lineas aut istorum
proprietatis, in quibus, ut voluit Aristoteles, mensura seu proportio perprius reperitur ....
Omnis igitur intensio successive acquisibilis ymaginanda est per rectam perpendiculariter
erectam super aliquod puctum aut aliquot puncta extensibilis spacii vel subjecti." Wieleit-
ner, "Ueber den Funktionsbegriff," p. 200.
86 "Tempus itaque sive duratio erit ipsius velocitatis longitudo et ejusdem velocitatis
intensio est sua latitude" Ibid., pp. 225-26. See also the diagrams in Tractatus de lati-
tudinibus formarum, fols. 202 r and 205 f .
87 Duhem has freely referred to Oresme as the inventor of analytic geometry. See, for
example, his article on Oresme in the Catholic Encyclopedia; cf . also his Etudes sur Uonard
de Vinci, HI, 386.
82 Medieval Contributions
notion that any geometric curve can be associated, through a co-
ordinate system, with an algebraic equation, and conversely. 88
However, Oresme's work marks a notable advance in mathematical
analysis in that it associated the study of variation with the repre-
sentation by coordinates. Although Aristotle had denied the existence
of an instantaneous velocity, the notion had continued to be invoked
implicitly, upon occasion, by Greek geometers and by Scholastic
philosophers. Oresme, however, was apparently the first to take the
significant step of representing an instantaneous rate of change by a
straight line. 89 He could not, of course, give a satisfactory definition
of instantaneous velocity, but he strove to clarify this idea by remark-
ing that the greater this velocity is, the greater would be the distance
covered if the motion were to continue uniformly at this rate. 90
Maclaurin, in attempting to clarify the Newtonian idea of a fluxion,
expressed himself very similarly almost four hundred years later, but
a rigorous and clear definition could be given only after the concept
of the derivative had been developed. Oresme, furthermore, was
confused by the perplexing problem of the indivisible and the con-
tinuum. In spite of his clear assertion that an instantaneous velocity
is to be represented by a straight line, he accepted the dictum of
Aristotle that every velocity persists throughout a time. 91 This view
implies a form of mathematical atomism which, although underlying
much of the thought leading to the calculus of Newton and Leibniz,
has been rejected by modern mathematics.
There is a widespread belief that the science of dynamics, which in
the seventeenth century played such a significant r61e in the formation
of the calculus, was almost entirely the product of the genius of
Galileo, who "had to create ... for us" 92 the "entirely new notion
... of acceleration." 93 That such a view is a gross misconception
will be clear to anyone who makes even a cursory examination of the
fourteenth-century doctrine of the latitude of forms. Oresme, for
88 Cf. Coolidge, "The Origin of Analytic Geometry," p. 233.
89 "Sed punctualis velocitas instantanea est ymaginanda per lineam rectam." Wieleit-
ner, "Ueber den Funktionsbegriff," p. 226.
90 "Verbi gratia; gradus velocitatis descencus est major, quo subjectum mobile magis
descendit vel descenderet si continuaretur simpliciter." Ibid., p. 224.
91 "Omnis velocitas tempore durat." Ibid., p. 225.
92 Mach, The Science of Mechanics, p. 133. See also Hogben, Science for the Citizen^
p. 241.
98 Mach, op. cit., p. 145.
Medieval Contributions
83
example, had a clear conception not only of acceleration in general
but also of uniform acceleration in particular. This is evident from
his statement that if the velocitatio (acceleration) is uniformis, then
the velocitas (velocity) is uniformiter dijformis; but if the velocitatio is
dijformis, then the velocitas is dijformiter dijformis. Oresme went
further and applied his idea of uniform rate of change and of graphical
representation to the proposition that the distance traversed by a
body starting from rest and moving with uniform acceleration is the
same as that which the body would traverse if it were to move for
the same interval of time with a uniform velocity which is one-half
the final velocity. Later, in the seventeenth century, this proposition
played a central r61e in the development both of infinitesimal methods
and of Galilean dynamics. It had been stated earlier and in a more
C
D
FIGURE 6
general form by Calculator, but Oresme and Galileo gave to it a geo-
metrical demonstration which, although not rigorous in the modern
sense, was the best which could be furnished before the integral
calculus had been established. 94
The proof of the theorem is based upon the fact that the motion
under uniform velocity, inasmuch as the latitude is the same at all
times, is represented by a rectangle such as ABGF (fig. 6) and that
the uniformly accelerated motion, in which the ratio of the change
in latitude to the change in longitude is constant, 95 corresponds to
94 In the mistaken belief that the work of Oresme preceded that of Calculator, Duhem
(Ittudes sur Leonard de Vinci, III, 388-98) has called this proposition the law of Oresme.
96 See Wieleitner, "Ueber den Funktionsbegriff," pp. 209-10; cf. also Traciatus de lati-
ttidinibus formarum, fol. 204 r . Duhem, upon the basis of these observations, has unwar-
rantedly asserted that Oresme "gives the equation of the right line, and thus forestalls
Descartes in the invention of analytical geometry." See his article, "Oresme," in the
Catholic Encyclopedia.
84 Medieval Contributions
the right triangle ABC. Oresme did not explicitly state the fact —
this is, of course, demonstrated in the integral calculus — that the areas
ABGF and ABC represent in each case the distance covered; but
this seems to have been his interpretation, 96 inasmuch as from the
congruence of the triangles CFE and EBG he concluded the equality
of the distances. 97 This is perhaps the first time that the area under
a curve was regarded as representing a physical quantity, but such
interpretations were to become before long commonplaces in the
application of the calculus to scientific problems. Oresme did not ex-
plain why the area under a velocity-time curve represented the
distance covered. It is probable, however, that he thought of the
area as made up of a large number of vertical lines or indivisibles,
each of which represented a velocity which continued for a very short
time. Such an atomistic interpretation is in harmony with the views
he expressed on instantaneous velocities and with the Scholastic
interest in the infinitesimal. Several centuries later this interpretation
was enunciated more boldly and vividly by Galileo, at a time when
atomistic conceptions enjoyed an even greater popularity in both
science and mathematics.
Arguments similar to those given by Oresme and Calculator ap-
peared also in the work of other men of the time, particularly at Oxford
and Paris. 98 William of Hentisbery, a famous logician at Oxford who
perhaps outdid Calculator in propounding sophisms on the subject of
motion, clearly stated the law for uniformly difform variation. 99
Marsilius of Inghen, at Paris, expounded this same law on the basis
of Oresme 's geometrical representation. 100 The traditions developed
at Oxford and Paris were continued also in the fifteenth century in
Italy, where Blasius of Parma (or Biagio Pelicani), reputed the most
versatile philosopher and mathematician of his time, explained the
law similarly, in his Questiones super tractatum de latitudinibus for-
marum. 101 This same principle for the uniformly difform acquisition
of qualities was known at Paris in the sixteenth century to Alvarus
96 Duhem, Etudes sur Leonard de Vinci, III, 394.
97 Wieleitner, "Ueber den Functionsbegriff," p. 230.
98 See Duhem, Etudes sur Leonard de Vinci, III, 405-81.
99 Wieleitner ("Der 'Tractatus'," pp. 130-32, n.) says that Hentisbury's arguments are
accompanied by diagrams, but Duhem (Eludes sur Leonard de Vinci, III, 449) says there
is no geometrical demonstration given with his writings.
h» Duhem, Etudes sur Leonard de Vinci, III, 399-405.
101 See Amodeo, "Appunti su Biagio Pelicani da Parma."
Medieval Contributions 85
Thomas, 102 John Major, Dominic Soto, and others. 103 The principles
of uniformly accelerated motion thus seem to have been common
knowledge to the Scholastics from the fourteenth century to the
sixteenth, and it is very probable that Galileo was familiar with their
work and made use of it in his development of dynamics. 104 At all
events, when the famous Discorsi of Galileo appeared in 1638, it
contained a diagram and a type of argument resembling strikingly
that previously given by Oresme. 105
The efforts toward the mathematical representation of variation
which resulted, in the fourteenth century, in the flurry of treatises on
the latitude of forms, were tied up in many ways with other ques-
tions related to the calculus. While he was discussing, in connection
with latitudo difformiter difformis, a form represented graphically by a
semicircle, the author of the Tractatus de latitudinibus formarum re-
marked that the rate of change of an intensity is least at the point
corresponding to the maximum intensity. 106 Great things have been
read into this casual remark, for historians of mathematics have
ascribed to its author the sentiments that the increment in the
ordinate of the curve is zero at its maximum point, and that the
differential coefficient vanishes at this point! 107 Ascribing such ideas
to the author of the Tractatus is obviously unwarranted, inasmuch as
they presuppose the clear conceptions of limit and differential quo-
tient which were not developed until many centuries later. 108 Further-
102 Wieleitner, "Zur Geschichte der unendlichen Reihen im christlichen Mittelalter,"
p. 154.
103 Duhem, Etudes sur Leonard de Vinci, III, 531 ff.
104 Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jakrhundert, pp. 243-44.
105 Mach (The Science of Mechanics, p. 131), probably unaware of the earlier work, has
ascribed the diagram and ideas to Galileo.
106 "In qualibet talis figura sua intensio terminatur ad summum gradum tarditatis et
sua remissio incipit a summo gradu tarditatis ut in medio puncto aliqualis ubi terminatur
intensio et incipit remissio, patet in figuris .c.d. et .d.c." Tractatus de latitudinibus forma-
rum, fol. 205 f . The figures referred to (fol. 204") are much like semicircles.
107 Curtze ("Ueber die Handschrift R. 4°, 2," p. 97) says that Oresme noticed in general
that "der Zuwachs .... der Ordinate einer Kurve in der unmittelbaren Nahe eines Maxi-
mums oder minimums gleich Null ist," and Moritz Cantor makes an even stronger state-
ment when he says (Vorlesungen, II, 120) "Oresme's Augen offenbarte sich die Wahrheit
des Satzes, den man 300 Jahre spater in die Worte kleidete, an den Hohen- undTiefpunkten
einer Curve sei der Differentialquotient der Ordinate nach der Abscisse Null."
108 Wieleitner ("Der 'Tractatus,' " p. 142) says "Ich halte aber die Bemerkung doch nur
fur rein anschauungsmassig"; and Timtchenko ("Sur un point du 'Tractatus de lati-
tudinibus formarum' de Nicholas Oresme") agrees, saying "qu'il 6tait encore assez loin du
theoreme exprimd par la formule — = (pour le sommet de la figure)."
dx
86 Medieval Contributions
more, the writer had apparently no idea of generalizing the statement
by extending the conclusion to other cases. His remarks do show
clearly, however, how fruitful the idea of the latitude of forms was to
be when it entered the geometry and algebra of later centuries, to
become eventually the basis of the calculus. The Scholastic philoso-
phers were striving to express their ideas in words and geometrical
diagrams, and were not so successful as we who realize, and can
make use of, the economy of thought which mathematical notation
affords.
Further consideration by Oresme of a latUudo dijjormiter difformis
led him, as it had also Suiseth somewhat earlier, to another topic
essential in the development of the calculus — that of infinite series.
In the same manner as Calculator, he considered a body moving
with uniform velocity for half a certain period of time, with double
this velocity for the next quarter of the time, with three times this
velocity for the next eighth, and so ad infinitum. He found in this
case that the total distance covered would be four times that covered
in the first half of the time. However, whereas Calculator had had
recourse to devious verbal argumentation in his justification of this
result, the method of Oresme was here (as in his earlier work) geo-
metrical and consisted in the comparison of the areas corresponding to
the distances involved in the graphical representation of the motion. 109
In this manner also he handled another similar case in which the
time was divided into parts j, ts, -is, . . . the velocity increasing in
arithmetic proportion as before. The total distance he found this time
to be -^ that covered in the first subinterval. Oresme then went on
to still more complicated cases. For example, he assumed that during
the first half of the time interval the body moved with uniform
velocity, then for' the next quarter of the time with motion uniformly
accelerated until the velocity was double the original, then for the
next eighth of the time uniformly with this final velocity, then for
the next sixteenth with uniform acceleration until the velocity was
again doubled, and so on. Oresme found that in this case the total
distance (or qualitas) is to that of the first half of the time as seven
is to two. 110 This is equivalent to summing geometrically to infinity
the series i + f + i+A + i+A + Tnr + A +
"• See Wieleitner, "Ueber den Funktionsbegriff," pp. 231-35.
«" Wieleitner, Ibid., p. 235.
Medieval Contributions 87
Calculator and Oresme were not the only Scholastics who were
concerned with such infinite series. In the anonymous tract written
before 1390, A est unum calidum, nx some of their results on series were
rediscovered. More than two centuries later, in 1509, similar work
appeared in the Liber de triplici motu of Alvarus Thomas, at Paris.
This book was intended to serve as an introduction to the Liber
calculationum of Suiseth and in it are handled several of the series
given earlier by Calculator and Oresme. However, the author went on
to similar but more complicated cases, in which he found (in the
manner of Calculator and Oresme) the sums to infinity of the series
t. T 4 2^8 2 2 ^ 16 2» ^ ' ' * ^3 2^6 2 a + 12 2*
-+- . . . , these sums being f and ^ respectively. 112 Alvarus Thomas
remarked that innumerable other such series can be found. In some
of these the sums involve logarithms, so that he could not arrive at
the sum exactly, but gave it approximately between certain integers.
Thus he said that the sum of 1 + -j • 5 + jj * 2 1 ~*~ 3 ' 2 1 "*~ ' * * ( wn i cn
results in 2 -f log 2) lies between 2 and 4. 113
It must be remembered that these infinite series were not handled
by the Scholastic philosophers as they are now in the calculus, for
they were given rhetorically, rather than by means of symbols, and
were bound up with the concept of the latitude of forms. Furthermore,
the results were found either through verbal arguments or geo-
metrically from the graphical representation of the form, rather than
by means of arithmetic considerations based upon the limit concept.
Had their work been more closely associated with the geometrical
procedures of Archimedes and less bound up with the philosophy of
Aristotle, it might have been more fruitful. It could then have antic-
ipated Stevin and the geometers of the seventeenth century in the
elimination, from the classical method of exhaustion, of the argu-
ment by a reductio ad absurdum, and in the substitution of an
analysis suggestive of the limit idea.
The fourteenth-century work at Paris and Oxford on the latitude
m Wieleitner, "Zur Geschichte der unendlichen Reihen," p. 167; cf. also Duhem,
tiudes sur Leonard de Vinci, III, 474-77.
m Wieleitner, "Zur Geschichte der unendlichen Reihen"; Duhem, Etudes sur Leonard
de Vinci, HI, 531 ff.
m Wieleitner, "Zur Geschichte der unendlichen Reihen," pp. 161-64; see also Duhem,
Ittudes sur Leonard de Vinci, HI, 540-41.
88 Medieval Contributions
of forms and related topics was not forgotten during the subsequent
decline of Scholasticism, for the doctrines spread throughout the
Italian universities. 114 In particular, they were taught at Pavia,
Bologna, and Padua by Blasius of Parma. 115 He, in turn, was referred
to a century later by Luca Pacioli and Leonardo da Vinci; and Nich-
olas of Cusa was very likely influenced by him. 116 The work of Calcu-
lator and Oresme was much admired by the men of the fifteenth and
sixteenth centuries, several printed editions of both the Liber cal-
culationum and the De latitudinibus formarum, as well as commen-
taries on each, appearing in the interval from 1477 to 1520. 117 In the
De subtilitate of Cardan, Calculator is classed among the great, along
with Archimedes, Aristotle, and Euclid. 118 Galileo, we have said, gave
in his dynamics a geometrical demonstration almost identical with
that of Oresme, and in another place referred to Calculator and
Hentisbery on activity in resisting media. 119 Descartes likewise, in
his attempt to determine the laws of falling bodies, employed argu-
ments closely resembling those of Oresme. 120 As late as the end of the
seventeenth century the reputation of Calculator was such that
Leibniz on several occasions referred to him as almost the first to
apply mathematics to physics and as one who introduced mathe-
matics into philosophy. 121
In spite of the reputation maintained by the great proponents of
the doctrine of the latitude of forms, however, the type of work they
represented was not destined to be the basis of the decisive influence
in the development of the methods of the calculus. The guiding
114 Duhem, Etudes sur Lionard de Vinci, III, 481-510.
115 See Thorndike, History of Magic, IV, 65-79.
116 See Amodeo, "Appunti su Biagio Pelicani da Parma," III, 540-53.
117 The Liber calculationum was published at Padua in 1477 and 1498, and again at
Venice in 1520; De latitudinibus formarum appeared at Padua in 1482 and 1486, at Venice
in 1505, and at Vienne in 1515.
™ Cardan, Opera, III, 607-8.
lu "Secunda dubitatio: quomodo se habent primae qualitates in activitate et resistentia.
De hac re lege Calculatorem in tractatu De reactione, Hentisberum in sophismate." See
Opere, I, 172.
™CEuvres,X, 58-61.
m "Quis refert, primum prope eorum, qui mathesin ad physicam applicarunt, fuisse
Johan. [sic] Suisset calculatorem ideo, scholasticis appellatum": Letter to Theophilus
Spizelius, April 7/17, 1670. Opera omnia (Dutens), V, 347. "Vellem etiam edi scripta
Suisseti, vulgo dicti calculatoris, qui Mathesin in Philosophicam Scholasticam introduxit,
sed ejus scripta in Cottonianis non reperio." Letter to Thomas Smith, 1696. Opera omnia
(Dutens), V, 567. See also Opera Omnia (Dutens), V, 421.
Medieval Contributions 89
principles were to be supplied by the geometry of Archimedes, al-
though these were to be modified by kinematic notions derived from
the quasi-Peripatetic disputations of the Scholastic philosophers on
the subject of variation. As early as the beginning of the fifteenth
century, the mensurational science and mathematics of Archimedes
were becoming more influential, so that men like Blasius of Parma in
some instances preferred his explanations to those of Aristotle. Blasius
of Parma is said to have written on infinitesimals 122 as well as upon
other mathematical topics, but no such works are known to be extant.
We do, however, have the work of Cardinal Nicholas of Cusa,
which illustrates the influence both of the Scholastic speculations
(perhaps through Blasius) and the work of Archimedes, as well as
another tendency which grew up, particularly on German soil, in the
fifteenth century. With the decline of the extremely rational and
rigorous thought of Scholasticism (the overpreciseness of which was
anathema to the growing Humanism of the time), 123 there was a trend
toward Platonic and Pythagorean mysticism. 124 This may well have
been in some measure responsible for the prevalence of mysticism 126
at the time. On the other hand, the development of science in Italy,
culminating in the dynamics of Galileo, owed much to the mathe-
matical philosophy of Plato and Pythagoras. 126 From the point of
view of the rise of the calculus, this pervasive Platonism exerted for
several centuries a not altogether unfortunate influence upon mathe-
matics, for it allowed to geometry what Aristotle's philosophy and
Greek mathematical rigor had denied — the free use of the concepts
of infinity and the infinitesimal which Platonic and Scholastic phi-
losophers had fostered. Nicholas of Cusa was not a trained mathe-
matician, but he was certainly well acquainted with Euclid's Elements
and had read Archimedes' works. Although he was primarily a theo-
logian, mathematics nevertheless constituted for him, as for Plato,
m Hoppe ("Geschichte der Infinitesimalrechnung," p. 179) says "Er ist also in dieser
Beziehung ein Vorlaufer von Cavalieri, von dem ihn 148 Jahre trennen." Blasius died in
1416, but probably Hoppe has in mind the 1486 edition of Blasius' Questiones, which is
separated from Cavalieri's Geometria in&ivisibilibus of 1635 by about 148 years.
123 Thorndike, History of Magic, III, 370.
m Lasswitz, Geschichte der Atomistik, I, 264-65; see also Burtt, The Metaphysical Founda-
tions of Modern Physical Science, pp. 18-42.
m Cf. Thorndike, Science and Thought in the Fifteenth Century, p. 22.
126 Strong, Procedures and Metaphysics, pp. 3-4; see also Cohn, Geschichte des Unend-
lichkeitsproblems, p. 95.
90 Medieval Contributions
the basis of his whole system of philosophy. 127 For him mathematics
was not restricted, as for Aristotle, to the science of quantity, but was
the necessary form for an interpretation of the universe. 128 Aristotle
had insisted on the operational character of mathematics and had
rejected the metaphysical significance of number, but Cusa revived
the Platonic arithmology. 129 He again associated the entities of mathe-
matics with ontological reality and restored to mathematics the cos-
mological status which Pythagoras had bestowed upon it. Further-
more, the validity of its propositions was regarded by him as established
by the intellect, with the result that the subject was not bound by the
results of empirical investigation.
This view of mathematics as prior to, or at least independent of,
the evidence of the senses encouraged speculation and allowed the
indivisible and the infinite to enter, so long as no inconsistency in
thought resulted. Such an attitude enriched the subject and eventu-
ated in the methods of the calculus, but this advantage was gained at
the expense of the rigor which characterized ancient geometry. The
logical perfection in Euclid found no counterpart in modern mathe-
matics until the nineteenth century. From the point of view of the
development of the calculus, this was not an unfortunate situation,
for the infinite and the infinitely small were allowed almost free reign,
even though it was about four hundred years before a logically satis-
factory basis for these notions was found and the limit concept was
definitely and rigorously established in mathematics.
In spite of Aristotle, the infinitely large and the infinitely small had
been employed somewhat surreptitiously by Archimedes in his geom-
etry, and the Scholastics had freely discussed these notions in their
dialectical philosophy; but with Nicholas of Cusa these ideas became,
together with an unnecessarily large admixture of Pythagorean and
theological mysticism, 130 a recognized part of the subject matter of
mathematics.
The definitions given by Nicholas of Cusa for the infinitely large
(that which cannot be made greater), and for the infinitely small
m Schanz, Der Cardinal Nicolaus von Cusa als Mothematiker, pp. 2, 7.
128 Lob, Die Bedeutung der Mathematik fur die Erkenntnislehre des Nikolaus von Kues,
pp. 36-37; cf. also Max Simon, Cusanus als Mothematiker.
»» Cf. L. R. Heath, The Concept of Time, p. 83.
180 Cohn, Geschichte des UnendlichkeitsproUems, p. 127.
Medieval Contributions 91
(that which cannot be made smaller) 131 are unsatisfactory, and his
mathematical demonstrations are not always unimpeachable. His
work is significant, however, in that it made use of the infinite and
the infinitesimal, not merely as potentialities, as had Aristotle, but as
actualities which are the upper and lower bounds of operations upon
finite magnitudes. Just as the triangle and the circle were to Nicholas
of Cusa the polygons with the smallest and the greatest number of
sides, so also zero and infinity were to him the lower and upper bounds
of the series of natural numbers. 132 His view was colored, moreover,
by his favorite philosophical doctrine — derived, perhaps, from the
Scholastic quasi-theological disputations on the infinite — that a finite
intelligence can approach truth only asymptotically. Consequently,
the infinite was the source and means, and at the same time the
unattainable goal, of all knowledge. 133 Cusa was led by this attitude
to view the infinite as a terminus ad quern to be approached only by
going through the finite 134 — an idea which represents a striving' toward
the limit concept 135 which was to be developed during the next five
hundred years.
Upon the basis of this view, Nicholas of Cusa proposed a char-
acteristic quadrature of the circle. If, contrary to Greek ideas of
congruence, the circle belongs among the polygons as the one with
an infinite number of sides and with apothem equal to the radius, its
area may be found by the same means as that employed for any
other polygon: by dividing it up into a number (in this case an infinite
number) of triangles, 136 and computing the area as half the product
of the apothem and the perimeter. Nicholas of Cusa added to this
explanation of the measurement of the circle an Archimedean proof,
using inscribed and circumscribed polygons, and a proof by the
reductio ad absurdum; but when his method was used later by Stevin,
Kepler, and others, the Archimedean proof was abandoned, an ele-
mentary equivalent of the limit concept being considered sufficient.
As Plato had opposed the Democritean atomic doctrine and yet
attempted to link the continuum with indivisibles, so also Cusa ob-
jected to Epicurean atomism, but at the same time regarded the line
131 Lorenz, Das Unendliche bet Nicolaus von Cues, p. 36. u * Lorenz, op. cit., p. 2.
133 Cohn, op. cit., p. 87. m Lasswitz, Geschichteder Atomistik, I, 282.
135 Cf. Simon, "Zur Geschichte und Philosophie der Differentialrechnung," p. 116.
136 Schanz, Nicolaus von Cusa, pp. 14r-15.
92 Medieval Contributions
as the unfolding of a point. In a similar manner, also, he held that
while continuous motion is thinkable, in actuality it is impossible, 137
inasmuch as motion is to be regarded as composed of serially ordered
states of rest. 138 These views call to mind the modern mathematical
continuum and the so-called static theory of the variable, but to
Nicholas of Cusa must not be ascribed any of the precision of state-
ment which now characterizes these matters. He did not make clear
in what manner the transition from the continuous to the discrete
is made. He asserted that although in thought the division of con-
tinuous magnitudes, such as space and time, may be continued indefi-
nitely, nevertheless in actuality this process of subdivision is limited
by the smallness of the parts obtained — that is, of the atoms and
instants. 139
The looseness of expression found toward the middle of the fifteenth
century in the formulation by Nicholas of Cusa of his views as to the
nature of the infinitesimal and the infinitely large was paralleled about
two centuries later by a similar lack of clarity in the multifarious
methods of indivisibles to which they to some extent gave rise. These
latter procedures, in turn, led to the differential and the integral cal-
culus. It is not to be concluded, however, that the views of Cusa are
to be construed as indicating any renaissance in mathematics, nor as
heralding the rise of the new analysis. The cardinal, absorbed as he
was in matters concerning church and state, did not contribute any
work of lasting importance in mathematics. Regiomontanus was per-
fectly justified in his criticism 140 of the repeated attempts which were
made by Cusa to square the circle. Nevertheless, as Plato's thought
may have been instrumental in the use of infinitesimals made by
Archimedes in his investigations preliminary to the application of the
rigorous method of exhaustion, so also the speculations of Nicholas
of Cusa may well have induced mathematicians of a later age to em-
ploy the notion of the infinite in conjunction with the Archimedean
demonstrations.
Leonardo da Vinci, who was strongly influenced, through Nicholas
137 L. R. Heath, The Concept of Time, p. 82.
138 Cohn, Geschichte des Unendlichkeits problems, p. 94.
13 » L. R. Heath, The Concept of Time, p. 82; Cohn, op. cit., p. 94; Lasswitz, Geschichte der
Atomistik, II, 276 ff.
140 See Kaestner, Geschichte der Matltematik, I, 572-76.
Medieval Contributions 93
of Cusa, by Scholastic thought, 141 and who was acquainted also with
the work of Archimedes, is said to have employed infinitesimal consid-
erations in finding the center of gravity of a tetrahedron by thinking
of it as made up of an infinite number of planes. We cannot, how-
ever, be sure of his point of view. 142 Cusa's ideas were more clearly
expressed in the work of Michael Stifel, who held that a circle may
correctly be described as a polygon with an infinity of sides, and that
before the mathematical circle there are all the polygons with a
finite number of sides, just as preceding an infinite number there are
all the given numbers. 143 Somewhat later Francois Viete also spoke
of a circle as a polygon with an infinite number of sides, 144 thus show-
ing that such conceptions were widely held in the sixteenth century.
The fullest expression of Nicholas of Cusa's mathematical thoughts
on the infinite and the infinitesimal, however, are found in the work
of Johann Kepler, who was strongly influenced by the cardinal's
ideas — speaking of him as divinus mihi Cusanus — and who was like-
wise deeply imbued with Platonic and Pythagorean mysticism. It
was probably the imaginative use by Cusa of the concept of infinity
which led Kepler to his principle of continuity — which included
normal and limiting forms of a figure under one definition, and in
accordance with which the conic sections were regarded as con-
stituting a single family of curves. 145 Such conceptions often led to
paradoxical results, inasmuch as the notion of infinity had not yet
received a sound mathematical basis; but Kepler's bold views suggested
new paths which were be to very fruitful later. Kepler lived a century
and a half after the time of Cusa. Moreover, unlike his predecessor,
although his early training was in theology, he was primarily a mathe-
matician. He was thus able, to the fullest extent, to take advantage
141 Duhem, Etudes sur Leonard de Vinci, II, 99 ff .
142 Libri, Histoire des sciences mathematiques en Ilalie, III, 41 ; Duhem, Etudes sur Lionard
de Vinci, I, 35-36. The former asserts that Leonardo used indivisibles; the latter questions
this. Duhem is wrong when he says in this connection (cf . Les Origines de la statique, II,
74) that Archimedes had restricted himself to plane figures in his work on the center of
gravity, since in the Method the centers of gravity of segments of a sphere and of a parabo-
loid are determined.
143 Cf. Gerhardt, Gesckichte der Mathematik in Deutschland, pp. 60-74.
144 Moritz Cantor, Vorlesungen, II, 539-40.
145 This is indicated by his statement, ". . . dico, lineam rectam esse hyperbolarum
obtussissimam. Et Cusanus infinitum circulum dixit esse lineam rectam. . . . Plurima
talia sunt, quae analogia sic vult efferi, non aliter." Kepler, Opera omnia, II, 595.
94 Medieval Contributions
of the accessibility of Archimedes' works resulting from their transla-
tion and publication during the middle of the sixteenth century.
It has been remarked that the medieval period added little to the
classical Greek works in geometry or to the theory of algebra. Its
contributions were chiefly in the form of speculations, largely from
the philosophical point of view, on the infinite, the infinitesimal, and
continuity, as well as of new points of view with reference to the
study of motion and variability. Such disquisitions were to play a
not insignificant part in the development of the methods and con-
cepts of the calculus, for they were to lead the early founders of the
subject to associate with the static geometry of the Greeks the graph-
ical representation of variables and the idea of functionality.
Both Newton and Leibniz, as well as many of their predecessors,
sought for the basis of the calculus in the generation of magnitudes —
a point of view which may be regarded as the most notable contribu-
tion of Scholastic philosophy to the development of the subject.
However, the sound mathematical basis for the seventeenth-century
elaboration of the infinitesimal procedures into a new analysis was
supplied by the mensurational treatises of Archimedes which had been
composed much earlier. These had become scattered and some had
been lost, although a number of them were familiar to the Arabs, and
although Archimedean manuscripts were known in the Scholastic
period. 146 No significant additions to his results were made until after
the appearance of printed editions of his treatises. In 1543 the Italian
mathematician, Nicolo Tartaglia, published at Venice portions of
Archimedes' work, including that on centers of gravity, the quadrature
of the parabola, the measurement of the circle, and the first book
on floating bodies. The editio princeps of Venatorius appeared at Basel
in 1544, and in 1558 the important translation by Federigo Com-
mandino appeared at Venice. Commandino himself composed a
treatise in the Archimedean manner — Liber de centro gravitatis soli-
dorum — showing that the study of the methods of the great Syracusan
mathematician had advanced to the point where new contributions
could be made. Shortly after this, however, numerous attempts were
made by Kepler and others to replace the almost tediously rigorous
148 See Thorndike and Kibre, Catalogue of Incipits, for citations of some dozen manu-
scripts of the twelfth, thirteenth, fourteenth, and fifteenth centuries.
Medieval Contributions 95
arguments of Archimedes by new methods which should be equivalent
to the older ones but at the same time possess a simplicity and an
ease of application to new problems which was lacking in the method
of exhaustion.
It is precisely the search for such substitutes which was to lead
within the next century, with the help of the Scholastic speculations,
to the methods of the calculus. For the sake of convenience and
unity, we shall make these anticipations the subject of the following
chapter.
IV. A Century of Anticipation
THE DEVELOPMENT of the concepts of the calculus may be
considered to have begun with the Pythagorean effort to com-
pare — through the superposition of geometrical magnitudes — lengths,
areas, and volumes, in the hope of thus associating with each configura-
tion a number. Thwarted in this by the problem of the incommensur-
ability of such magnitudes, it was left for later Greek geometers to
circumvent such direct comparison through the method of exhaustion
of Eudoxus. Through this method the need for the infinite and the
infinitesimal had been obviated, although such notions had been con-
sidered by Archimedes as suggestive heuristic devices, to be used in
the investigation of problems concerning areas and volumes which
were preliminary to the intuitively clear and logically rigorous proofs
given in the classical geometrical method of exhaustion.
During the later Middle Ages the conceptions of the infinite and
the infinitesimal, and the related ideas of variation and the continuum,
were more freely discussed and utilized. Because, however, the in-
terest in such s peculations remained less closely associated with geom-
etry than with metaphysics or science, or even with attempts to explain
the possibility of natural magic, 1 the primary influence leading to the
calculus was n^t supplied by the doctrines of the Scholastic philoso-
phers but by 1he greater enthusiasm for the methods of Archimedes
which was mar if est in the late sixteenth century and which continued
throughout the whole of the century following. Whereas a multiplicity
of printed editi :>ns of the works of Jordanus Nemorarius, Bradwardine,
Calculator, Orcsme, Hentisbery, and other medieval scholars had ap-
peared in the ]ate fifteenth and early sixteenth centuries, there arose
toward the mic die of the latter century a strong opposition, illustrated
by the attitude of Ramus, to Aristotelianism and Scholastic method-
ology. 2 It was during the height of this reaction that the works of
Archimedes ap reared in numerous editions and, admiring Archimedes,
1 Thorndike, Hi %tory of Magic, III, 371.
2 Johnson and Larkey, "Robert Recorde's Mathematical Teaching and the Anti-
Aristotelian Mover lent."
A Century of Anticipation 97
the men of the time refused to recognize the work of the Middle
Ages. 3
In the mathematics of this period, nevertheless, the intrusion of in-
fluences from the Scholastic age is easily discernible in the attempts to
reconcile with the thought of Archimedes the newly discovered infin-
itesimal methods which were fostered. There is noticeable in this de-
velopment, as well, a tendency foreign alike to Archimedes and to the
Scholastic period — a deeper interest in the Arabic algebra which had
been developing in Italy, in which the concept of infinity did not
figure. The algebra of Luca Pacioli's Summa de arithmetica was not
greatly advanced over that to be found in Leonardo of Pisa almost
three hundred years earlier, 4 but in the sixteenth century the subject
was assiduously studied again. Before 1545 the cubic had been solved
by Tartaglia and Cardan, and the quartic by Ferrari; and thereafter
a freer use of irrational, negative, and imaginary numbers was made
by Cardan, Bombelli, Stifel, and others. The Greeks had not regarded
irrational ratios as numbers in the strict sense of the word, and the
attitude in the medieval period had been similar. Bradwardine as-
serted that an irrational proportion is not to be represented by any
number; 5 and Oresme, in discussing the popular question of whether
the celestial motions were commensurable or incommensurable,
concluded that geometry favored the latter but arithmetic the
former. 6
The Hindus and the Arabs, on the other hand, had not clearly dis-
tinguished between rational and irrational numbers. On adopting the
Hindu-Arabic algebra, the sixteenth-century mathematicians had
continued to employ the irrational ratio. They now recognized this as
a number, but they stigmatized it, following Leonardo of Pisa, as a
numerus surdus, and they continued to interpret it geometrically as a
ratio of lines. 7 Negative quantities, admitted by the Hindus but not
by the Greeks or the Arabs, were accepted in the sixteenth century as
numeri falsi or ficti, but in the following century these were recog-
3 Duhem, Les Origines de la statique, I, 212. 4 Cajori, History of Mathematics, p. 128
5 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 158.
• Thorndike, History of Magic, III, 406.
7 See Pringsheim, "Nombres irrationnels et notion de limite," pp. 137-40; Fink, A Brief
History of Mathematics, pp. 100-1. However, Kepler, as late as 1615 spoke of the irrational
as "ineffable." Opera omnia, IV, 565.
98 A Century of Anticipation
nized as numbers in the strict sense of the word. 8 Imaginary numbers
also were regularly employed after the sixteenth century, although
they continued to occupy an anomalous position in mathematics until
the time of Gauss.
Such generalizations of number, although not at the time based
upon satisfactory definitions, were influential later in leading to the
limit concept and to the arithmetization of mathematics. More im-
portant than this, in the development of the algorithm of the calculus,
was the systematic introduction, during the later sixteenth century, of
symbols for the quantities involved in algebraic relations.
As early as the thirteenth century, letters had been used as symbols
for quantities by Jordanus Nemorarius in his science and mathematics.
Their establishment as symbols of the abstract quantities entering into
algebra, however, was largely the work of the great French mathe-
matician Francois Viete, 9 who used consonants to represent known
quantities and vowels for those unknown. He distinguished arith-
metic, or logistica numerosa, from algebra, or logistica speciosa, thus
making the latter a calculation with letters rather than with numbers
alone.
This literal symbolism was absolutely essential to the rapid progress
of analytic geometry and the calculus in the following centuries, 10 for
it permitted the concepts of variability and functionality to enter into
algebraic thought. The improved notation led also to methods which
were so much more facile in application than the cumbrous geometrical
procedures of Archimedes, of which they were modifications, that
these methods were eventually recognized as forming a new analysis
— the calculus. The period during which this transformation took place
may be considered as the century preceding the work of Newton and
Leibniz. 11
Numerous translations of the works of Archimedes had been made
during the middle of the sixteenth century, and soon after this mathe-
8 See Fine, Number System, p. 113; Paul Tannery, Notions historiques, pp.,333-34; Fehr,
"Les Extensions de la notion de nombre dans leur developpement logique et historique."
9 Cajori, History of Mathematics, p. 139.
10 Cf. Karpinski, "Is There Progress in Mathematical Discovery?" p. 47.
11 For a comprehensive account of the methods developed during this period, see
Zeuthen, Geschichte der Mathcmatik itn -XVI. una 1 XVII. Jahrhundert. Zeuthen's account
is an outline of the development of the subject in this interval, rather than of fundamental
concepts, and so includes a large amount of mathematical detail.
A Century of Anticipation 99
maticians had reached the point at which original contributions could
be made to the classical work of the Greeks. This is evidenced by the
fact that Commandino in 1565 published a work of his own on centers
of gravity. In this he proved, among other things, that the center of
gravity of a segment of a paraboloid of revolution is situated on the
axis two-thirds of the distance from the vertex to the base. 12 This was
a proposition which Archimedes had, incidentally, demonstrated by
infinitesimals in a treatise, the Method, which was apparently not
known at that time. Commandino 's proof followed the orthodox style
of the method of exhaustion. Such extensions, however, had perhaps
less influence on the development of the calculus than did certain
striking innovations in method introduced by succeeding mathe-
maticians. Perhaps the first of such significant modifications is that
advanced by Simon Stevin of Bruges in 1586, very nearly a century
before the first printed work on the calculus by Leibniz, in 1684, and
that of Newton in 1687.
Stevin was essentially an engineer and a practical-minded scientist.
For this reason he had perhaps less regard for the philosophy of science
and the exigencies of mathematical rigor than for technological tradi-
tion and methodology. 13 As a result, Stevin did not merely imitate, as
had Commandino, Archimedes' use of the method of exhaustion: he
accepted the direct portion of his characteristic proof as sufficient to
establish the validity of any proposition that required it, without add-
ing in every case the formal reductio ad absurdum required by Greek
rigor. Furthermore, he frequently omitted, as we do in the integral
calculus, one of the approximating figures which Archimedes had used,
being satisfied with either the inscribed or the circumscribed figure
only. Stevin demonstrated as follows (in his work on statics, in 1586)
that the center of gravity of a triangle lies on its median. Inscribe in
the triangle ABC a number of parallelograms of equal height, as il-
lustrated (fig. 7). The center of gravity of the inscribed figure will lie
on the median, by the principle that bilaterally symmetrical figures
are in equilibrium (a principle used by Archimedes in proving the law
of the lever, and also by Stevin in his well-known demonstration of
the law of the inclined plane). However, we may inscribe in the tri-
a Commandino, Liber de centro gravitatis solidorum, fol. 40"-41 B .
u Strong, Procedures and Metaphysics, pp. 91-113.
100
A Century of Anticipation
angle an infinite number of such parallelograms, for all of which the
center of gravity will lie on AD. Moreover, the greater the number of
parallelograms thus inscribed, the smaller will be the difference be-
tween the inscribed figure and the triangle ABC. If, now, the "weights"
of the triangles ABD and ACD are not equal, they will have a certain
fixed difference. But there can be no such difference, inasmuch as each
of these triangles can be made to differ by less than this from the sums
of the parallelograms inscribed within them, which are equal. There-
fore the "weights" of ABD and ACD are equal, and hence the center
of gravity of the triangle ABC lies on the median AD. U
Exactly analogous demonstrations were given by Stevin of propo-
sitions on the centers of gravity of plane curvilinear figures, including
the parabolic segment. These proofs given by Stevin indicate the direc-
tion in which the method of limits was developed as a positive con-
cept. The Greek method of exhaustion had not boldly concluded, as
did Stevin, that inasmuch as the difference could, by continued sub-
division, be shown to be less than any given quantity, there could as
a result be no difference. The Greeks felt constrained in every case to
add the full reductio ad absurdum proof to show the equality. Stevin
did not, of course, speak of the triangle as the limit of the sum of the
14 Stevin, Hypomnemata mathematica, IV, 57-58; cf . also Bosnians, "Le Calcul infinitesi-
mal chez Simon Stevin."
A Century of Anticipation
101
inscribed parallelograms; but it would require only slight changes in
his method — largely in the nature of further arithmetization and the
use of greater precision in terminology — to recognize in it our modern
method of limits.
That Stevin regarded his approach to these problems as a significant
modification of the method of Archimedes may be indicated in his
proof of the proposition on the center of gravity of a conoidal segment
— a proposition "the demonstration of which the ingenious and subtle
mathematician Fredericus Commandinus gives in proposition 29 de
solidorum centrobaricis, and which is arranged as follows in accordance
D
FIGURE 8
with our custom and method." 15 Circumscribe about the segment
ABC two cylindrical segments FGBC and MLIK as illustrated (fig. 8).
Now the centers of gravity of these cylindrical segments are, by the
principle of the equilibrium of bilaterally symmetrical figures, at the mid-
points, N and O, of their axes, AH and HD, respectively; and the
center of gravity of the entire circumscribed figure is at R such that
NR = 2RO. Letting E be the point such that AE = 2ED, it can be
shown that ER = tsAD. If, now, one similarly circumscribes about
15 Stevin, Hypomnemata mathematica, II, 75-76.
102 A Century of Anticipation
the segment ABC four such cylindrical segments of equal height, the
center of gravity of this circumscribed figure is found to lie above E at
a point L such that EL = -riAD. Successively doubling the number of
these cylinders, the center of gravity of the circumscribed figure re-
mains always above E and will differ from E by teAD, tsAD, and so
on. Thus the center of gravity descends, approaching E more and more
closely. Modern mathematics would now conclude that E is the limit
of the center of gravity of the circumscribed figure and therefore is the
center of gravity of the conoidal segment. Stevin, however, reached
this conclusion only after cautiously observing that similarly the center
of gravity of the analogous inscribed figure ascended toward E in the
same manner.
The demonstration given by Stevin, in the above proposition in
terms of the sequence ts, ih, tV, sV, • . ., is comparable to that em-
ployed by Archimedes in his quadrature of parabola, in which the
series 1 + £ + T \ + ^ + . . . figured. Both the Greek mathemati-
cian and the Flemish engineer, in their use of such sequences and
series, stopped short of the limit concept. Neither thought of such a
sequence or series as carried out to an infinite number of terms in the
modern sense. Archimedes explicitly stopped with the nth term j^
and added the term, ^ • 7—^, representing the remainder of the series;
Stevin used the word infinite in the Peripatetic sense of potentiality
only — the sequence could be continued as far as desired, and the error
consequently made as small as one pleased. For this reason Archimedes
had been constrained to complete his work through the demonstration
by a reductio ad absurdum; Stevin, although he adduced no such for-
mal argument, had recourse also to supplementary demonstrations,
such as the inclusion, in the above proposition on the conoid, of a
second sequence approaching the point from the other side. This hesi-
tation on the part of Stevin to accept as sufficient the notion of the
limit of an infinite sequence is apparent also in a proposition in his hy-
drostatics, which represented perhaps his nearest approach to the
method of limits. Here he supplemented the "mathematical demon-
stration" of propositions, carried out as above, by a "demonstration
by numbers," suggested, perhaps, by the then recent Italian work in
algebra and mensuration, and encouraged by the neglect of geometry
A Century of Anticipation 103
in favor of arithmetic found in the Netherlands during the sixteenth
century. 16
In supplementing the proof 17 that the average pressure on a vertical
square wall of a vessel full of water corresponds to the pressure at its
mid-point, he gave an "example with demonstration." He subdivided
the wall into 4 horizontal strips and noted that the force on each is
greater than 0, tV, ts, and ts units, and less than t*, te, A, and tt
units respectively; so that the total force is greater than ts and less
than H- If the wall is subdivided into 10 horizontal strips, the force is
found similarly to be greater than tou and smaller than rh units; on
using 1,000 strips, it is determined as more than x^V^oo un i ts > an d
less than ^""^oVo units. 18 By increasing the number of strips, he then
remarked, one may approach as closely as desired to the ratio one-half,
thus proving that the force corresponds to that which would be ob-
served if the wall were placed horizontally at a depth of half a unit. 19
This "demonstration by numbers" would correspond exactly to that
given in the calculus if Stevin had limited himself to one of his two
sequences and had thought of the results given by his successive sub-
divisions of the wall as forming literally an infinite sequence with the
limit \. Stevin, however, shared not only the Greek view with regard
to infinity but also, although to a lesser extent than did most of his
mathematical contemporaries, the classical apotheosis of geometry, so
characteristic of ancient mathematics, which was to prevail in the
seventeenth century. Even he considered his arithmetic proof, outlined
above, as merely a mechanical illustration to be distinguished from the
general mathematical demonstration. 20
Perhaps the one tendency that did more than any other to conceal
from mathematicians for almost two centuries the logical basis of the
16 Cf. Struik, "Mathematics in the Netherlands during the First Half of the XVIth
Century."
17 Stevin, Hypomnemata mathetnatica, II, 121 ff.
a It will be recalled that Stevin had introduced his use of decimal fractions, in De
Thiende, which had appeared in 1585, the year before the publication of his work in statics.
See Sarton, "The First Explanation of Decimal Fractions and Measures (1585)."
19 Stevin, Hypomnemata mathetnatica, II, 125-26; cf. also Bosnians, "Le Calcul chez
Stevin," pp. 108-9.
20 "Mathematicae & mechanicae demonstrationis a doctis annotatur differentia, neque
injuria. Nam ilia omnibus generalis est, & rationem cur ita sit penitus demonstrat, haec
vero in subjecto duntaxat paradigmate numeris declarat." Stevin, Hypomnemata mathe-
tnatica, II, 154.
104 A Century of Anticipation
calculus was the result of the attempt to make geometrical, rather
than arithmetic, conceptions fundamental. This will be more true of
Stevin's successors than it was of him. It must be borne clearly in
mind, however, that although the logical basis of the calculus is arith-
metic, the new analysis resulted largely from suggestions drawn from
geometry.
The procedures substituted by Stevin for the method of exhaustion
constituted a marked step toward the limit concept. The extent of his
influence on contemporary thought, however, is difficult to determine.
The work in statics which contains his anticipations of the calculus ap-
peared in Flemish in 1586. It was included also in the Flemish, French,
and Latin editions of his mathematical works published in 1605-8, and
in a later French translation of his works by Girard in 1634. 21 With the
exception of the last-mentioned edition, however, most of these were
not easily accessible to mathematicians, 22 and, by the time of the French
translation of 1634, there had already appeared in Italy and Germany
a number of alternative methods which were destined to become much
more widely known than those of Stevin. Nevertheless, the influence
of the Flemish scientist is evident in the thought of a number of later
mathematicians of the Low Countries. 23 Before we turn to these men,
however, it may be well to indicate briefly the nature of the modifica-
tions of Archimedes' work which appeared in Italy and Germany
shortly after Stevin had published his methods.
Luca Valerio likewise attempted, in his De centro gravitatis solidorum
of 1604, a methodization of Archimedes' procedure which should ob-
viate the need for the reductio ad absurdum and yet retain the neces-
sary rigor of demonstration. His change was not so sweeping as that of
Stevin, and his view is not so closely related to the modern. He merely
tried to substitute for the method of Archimedes a few general theorems
which could be cited instead of carrying through the details of proof in
each and every case. No attempt had been made in Greek geometry to
establish such propositions which might simply be quoted in particular
cases in lieu of the double reductio ad absurdum demonstration. 24
a See the article by Sarton, "Simon Stevin of Bruges (1548-1620)"; and that by Bos-
mans, "Simon Stevin," for biographies of Stevin, analyses of his work, and extensive
bibliographical references.
22 Bosnians, "Simon Stevin," p. 889.
a Bosnians, "Sur quelques exemples de la mSthode des limites chez Simon Stevin."
u The Works of Archimedes, Introduction, p. cxliii.
A Century of Anticipation
105
Valerio professed that it was the appearance of Commandino's work
on centers of gravity which had encouraged him, 25 as it had Stevin, 26
to attempt such a modification of the method of Archimedes. The sig-
nificant generalization is found in the proposition that, given any figure
in which the distance between points on opposite sides of a diameter
vanishes, parallelograms can be inscribed and circumscribed in such a
way that the excess of the circumscribed figure over that inscribed is
less than any given area. 27 This is proved by observing that the excess
is in each case equal to the area of the parallelogram BF (fig. 9) and
that for suitably chosen approximating figures this "is less than a given
area." 28 Then Valerio assumed without proof that if the difference be-
FIGURE 9
tween the inscribed and the circumscribed figures is smaller than any
given area, this will be true also of the difference between the curve
and either of these figures. This geometrical reasoning is strikingly
similar to that presented in many present-day elementary textbooks
on the calculus. Valerio did not, however, regard the area of the curve
as necessarily defined by the limit of the area of either the inscribed
or the circumscribed figure, as the number of such parallelograms be-
comes infinite. This is a sophisticated arithmetical conception which
25 Valerio, De centra gravitalis solidorum libri tres, p. 1.
26 It is doubtful whether Valerio knew of the work of Stevin. See Bosnians, "Les Demon-
strations par l'analyse mfinitesimale chez Luc Valerio," p. 211.
27 Valerio, De centra gravitotis, p. 13.
28 "Sed parallelogrammum BF est minus superficie proposita." Ibid., p. 14.
106 A Century of Anticipation
was not established until two centuries later. Nevertheless, Valerio was
in a sense anticipating the limit concept, in geometrical form, to the
extent of indicating the necessary condition for the existence of such a
limit — viz., that the difference in these areas can be made less than
any specified area.
Having generalized, by the above demonstration, the method of in-
scribing areas, Valerio then proceeded to state general propositions
which should replace, at least in so far as ratios were concerned, the
argument by a reductio ad absurdum as found in the method of ex-
haustion. The intention of these may be stated in the following form:
If four magnitudes, A, B, C, D, are given, and if two others, G and H,
can be found which are at the same time greater or smaller than A and
. G .
B by a magnitude less than any given magnitude, and if the ratio — is
H
at the same time greater or smaller than the ratios — and — , then —
5 B D B
C
= — . 29 This proposition is significant not only as a methodization of
Archimedes' procedure, but also as a vague striving toward the idea
which is now expressed concisely by saying that the limit of a ratio of
two variables is equal to the ratio of the limits of these variables. This
latter concept is dependent, however, on considerations of the infinite
— on speculations in which Valerio did not indulge, but which inter-
ested many of his contemporaries and successors, particularly those
who combined mathematical investigations with theological interests.
Johann Kepler had been educated with the intention of entering the
Lutheran ministry, but had been forced to turn later to the teaching of
mathematics as a means of earning his livelihood. This may account in
part for the fact that, although he owed as much to Archimedes as had
Stevin, the nature of his work is so markedly different from that of the
engineer of Bruges. There is in the thought of Kepler a deep strain of
mysticism which was lacking in the attitude of Stevin, of whom Kepler
may have known. 30 However, this speculative tendency on the part of
11 De centro gravitatis, p. 69; see also Wallner, "Grenzbegriffes," p. 251.
30 Bosnians ("Les D6monstrations sur l'analyse infinitesimale chez Luc Valerio," p. 211)
holds that Kepler was evidently familiar with Stevin's work; and Hoppe ("Zur Geschichte
der Infinitesimalrechnung," p. 160) that he may have known of it; Wieleitner ("Das
Fortleben") believes that such an assumption is not justified.
A Century of Anticipation 107
Kepler is perhaps with more justice to be ascribed to the Platonic-
Pythagorean influence, which had been strong in Europe in the pre-
vious century, 31 and which was so evident in the thought of Cusa, to
whom Kepler owed at least part of his inspiration. 32 The belief that the
universe was an ordered mathematical harmony, so strongly shown in
Kepler's Mysterium cosmographicum, was combined with Platonic and
Scholastic speculations on the nature of the infinite, giving him a modi-
fication of Archimedes' mensurational work which was to be a power-
ful influence in shaping the development of the calculus.
The ancient Greek philosophers, in their search for unity in this uni-
verse of perplexing multiplicity, had failed for two reasons to bridge
the gap between the curvilinear and the rectilinear: first, they banned
the infinite from geometry; and second, they hesitated, following the
discovery of the irrational, to pursue further the Pythagorean asso-
ciation of numerical considerations with geometrical configurations.
The pious enthusiasm of Kepler, however, saw in this impasse but one
more evidence of the handiwork of the Creator, who had established all
things in harmony. God wished quantity to exist so that the comparison
between a curve and a straight line might be made. This fact was made
clear to him by the "divine Cusanus" and by others who had regarded
the forms of the curve and the straight line as complementary, daring
to compare the curve to God and the line to his creatures. "For this
reason those who have tried to relate the Creator to his handiwork and
God to man and divine judgments to human represent an occupation
which is by no means more useful than that of those who seek to com-
pare the circle with the square." 33 Under such inspiration, guided as
well by the speculations of Cusa and Giordano Bruno on the infinite in
cosmology, 34 and fortified by the knowledge "that nature teaches
geometry by instinct alone, even without ratiocination," 35 Kepler
was led to develop his modification of the procedures of Archi-
medes.
The task of writing a complete treatise on volumetric determina-
tions seems to have been suggested to Kepler by the prosaic problem
of determining the best proportions for a wine cask. The result was
31 Burtt, Metaphysical Foundations, pp. 44-52; Strong, Procedures and Metaphysics, pp.
164 ff.
32 Cf. Kepler, Opera omnia, I, 122; II, 490, 509, 595.
» Ibid., I, 122. M Ibid., II, 509. « Ibid., IV, 612.
108 A Century of Anticipation
the Nova stereometria, which appeared in 1615. 36 This contains three
parts, of which the first is on Archimedean stereometry, together with
a supplement containing some ninety-two solids not treated by Archi-
medes. The second part is on the measurement of Austrian wine barrels,
and the third on applications of the whole.
Kepler opened his work on curvilinear mensuration with the simple
problem of determining the area of the circle. In this he abandoned the
classical Archimedean procedures. He did not substitute for these the
limiting consideration proposed by Stevin and Valerio, but had re-
course instead to the less rigorous but more suggestive approach of
Nicholas of Cusa. Like Stifel and Viete, he regarded the circle as a
regular polygon with an infinite number of sides, and its area he there-
fore looked upon as made up of infinitesimal triangles of which the
sides of the polygon were the bases and the center of the circle the
vertex. The totality of these was then given by half the product of
the perimeter and the apothem (or radius). 37
Kepler did not limit himself to the simple proposition above, but
with skill and imagination applied this same method to a wide variety
of problems. By looking upon the sphere as composed of an infinite
number of infinitesimal cones whose vertices were the center of the
sphere and whose bases made up the surface, he showed that the vol-
ume is one-third the product of the radius and the surface area. 38 The
cone and cylinder he regarded variously: as made up of an infinite
number of infinitely thin circular laminae (as had Democritus two
thousand years earlier), as composed of infinitesimal wedge-shaped
segments radiating from the axis, or as the sum of other types of
vertical or oblique sections. 39 The volumes of these he computed by
the application of such views. In a similar manner he rotated a circle
about a line and calculated, by infinitesimal methods, the volume of
the anchor ring, or tore, thus generated. 40 This determination was
equivalent to an application, for a special case, of the classical theorem
of Pappus, later called Guldin's rule. Kepler then extended his work
to solids not treated by the ancients. The areas of the segments cut
from a circle by a chord he rotated about this chord, obtaining solids
36 Followed, about a year later, by a popular German edition. The Latin appears, with
notes, in Volume IV, Opera omnia; the German in Volume V.
37 Opera omnia, IV, 557-58.
» Ibid., IV, 563. 3 » Ibid., IV, 564, 568, 576 ff. « Ibid., IV, 575-76, 582-83,
A Century of Anticipation 109
which he designated characteristically as apple or citron-shaped, ac-
cording as the generating segment was greater or less than a semi-
circle. 41 The volumes of these and other solids he likewise calculated
by his infinitesimal methods. To Willebrord Snell, the editor of Stevin's
works, Kepler proposed, as a challenge, the determination of the solids
obtained similarly by the rotation of segments of conic sections. 42 This
problem was significant in the later work of the Italian mathematician
Cavalieri.
Some of Kepler's summations are remarkable anticipations of re-
sults found later in the integral calculus. 43 In his well-known Astro-
nomia nova of 1609, for example, there is a computation 44 resembling
that which is expressed in modern notation as Jo sin 6 dd = 1 — cos 6.
Other calculations in this work correspond to approximations to elliptic
integrals, 45 in one of which v(a + b) is given as the approximate length
of the ellipse with the semi-axes a and b. 46 Kepler, however, was far
from clear on the point of the basic conceptions involved, with the re-
sult that his work is not free from errors. 47 He generally spoke of sur-
faces and volumes as made up of infinitesimal elements of the same
dimension, but occasionally he lapsed into the language of indi-
visibles, which his successor Cavalieri was to find so congenial. In one
place he spoke of the cone as though composed of circles, 48 and in the
work by which he arrived in his astronomy at his famous second law
he regarded the sector of an ellipse as the sum of its radius vectors. 49
Kepler appears not to have distinguished clearly between proofs by
means of the method of exhaustion, by ideas of limits, by infinitesimal
elements, or by indivisibles. The conceptions which he held in his dem-
a Wolf (History of Science, Technology, and Philosophy in the Sixteenth and Seventeenth
Centuries, pp. 204-5) has correctly pointed out that the word citrium, which Kepler used
in this connection, is properly translated as "gourd"; but he has failed to add that Kepler
himself, in his German edition, translated it as "citron." Opera omnia, V, 526.
"Ibid., IV, 601,656.
43 See Struik, "Kepler as a Mathematician," in Johann Kepler, 1571-1630. This volume
also contains an excellent bibliography of Kepler's works, by F. E. Brasch.
44 See Opera omnia, III, 390; cf. also Gunther, "Uber ein merkwiirdige Beziehung
zwischen Pappus und Kepler"; Enestrom, "Uber die angebliche Integration einer trig-
onometrischen Funktion bei Kepler."
45 Cf. Struik, "Kepler as a Mathematician," p. 48; Zeuthen, Geschichte der Mathematik
im XVI. und XVII. Jahrhunderl, pp. 254-55.
46 Opera omnia, III, 401. 47 Moritz Cantor, Vorlesungen, II, 753.
48 "Nam conus est hie veluti circulus corporatus." Opera omnia, IV, 568.
49 Opera omnia, III, 402-3.
110 A Century of Anticipation
onstrations are a far cry from the notions held by the ancient geom-
eters. The Greek thinkers saw no way of bridging the gap between the
rectilinear and the curvilinear which would at the same time satisfy
their strict demands of mathematical rigor and appeal to the clear
evidence of sensory experience. Fortified by the scholastic disputations
on the categorical infinite and by Platonic mathematical speculations,
Kepler followed Nicholas of Cusa in resorting to a vague "bridge of
continuity" which finds no essential difference between a polygon and
a circle, between an ellipse and a parabola, between the finite and the
infinite, between an infinitesimal area and a line. 50 This striving for an
expression of the idea of continuity constantly reappears throughout
the period of some fifty years preceding the formulation of the methods
of the calculus. Leibniz himself, like Kepler, frequently fell back upon
his so-called law of continuity when called upon to justify the differ-
ential calculus; and Newton concealed his use of the notion of con-
tinuity under a concept which was empirically more satisfying, though
equally undefined — that of instantaneous velocity, or fluxion.
Kepler's Doliometria, or Stereometria doliorum, exerted such a strong
influence in the infinitesimal considerations which followed its ap-
pearance, and which culminated a half century later in the work of
Newton, that it has been called, with perhaps pardonable exaggeration,
the source of the inspiration for all later cubatures. 81 Before passing on
to these anticipations of the integral calculus, there should be cited a
contribution which Kepler made to the thought leading to the differ-
ential calculus. The subject of the measurement of wine casks had led
Kepler to the problem of determining the best proportions for these. 62
This brought him to the consideration of a number of problems on
maxima and minima. In the Doliometria he showed, among other
things, that of all right parallelepipeds inscribed in a sphere and having
square bases, the cube is the largest; 53 and that of all right circular
cylinders having the same diagonal, that one is greatest which has the
diameter and altitude in the ratio of V2 to l. 54
These results were obtained by making up tables in which were
60 Cf . Taylor, "The Geometry of Kepler and Newton."
61 Moritz Cantor, Vorlesungen, II, 750.
62 Kepler discovered, incidentally, that the Austrian barrels approximated very closely
the desired proportions.
» Opera omnia, IV, 607-9. M Ibid; IV, 610-12.
A Century of Anticipation 111
listed the volumes for given sets of values of the dimensions, and from
these selecting the best proportions. An inspection of such tables
showed him an interesting fact. He remarked that as the maximum
volume was approached, the change in the volume for a given change
in the dimensions became smaller. Oresme, several centuries earlier,
had made a similar observation, but had expressed it differently.
Oresme had noticed that for a form which was represented graphically
by a semicircle, the rate of change was least at the maximum point.
This thought appeared again, in the seventeenth century, in the
methods of the French mathematician Fermat. Whether Fermat was
influenced in this direction by Kepler or Oresme is problematical; but
a comparison of the distinctly different points of view which the latter
men represent will aid later in understanding Fermat's approach.
Kepler had made his remark upon the basis of numerical considera-
tions. He was, moreover, more particularly concerned with static con-
siderations as found in Greek geometry and in methods of indivisibles.
He consequently expressed himself in terms of increments and decre-
ments near the maximum point. On the other hand, the medieval
problem of the latitude of forms and the graphical representation of
continuous variability had led Oresme to state the conclusion in terms
of the rate of change. The latter view has been made fundamental in
mathematics through the concept of the derivative; but it is Kepler's
mode of expression which appeared in the work of Fermat. Although
the Scholastic views on variation played a significant role in the an-
ticipations of the calculus, the static approach of Kepler predominated.
Increments and decrements, rather than rates of change, were the
fundamental elements in the work leading to that of Leibniz, and
played a larger part in the calculus of Newton than is usually recog-
nized. The differential became the primary notion and it was not ef-
fectively displaced as such until Cauchy, in the nineteenth century,
made the derivative the basic concept.
Twenty years after the publication of the Stereometria doliorum of
Kepler there appeared in Italy a work which rivaled it in popularity.
So famous did the Geometria indivisibilibus of Bonaventura Cavalieri
become that it has been maintained, with some justice, that the new
analysis took its rise from the appearance, in 1635, of this book. 55 To
68 Leibniz, The Early Mathematical Manuscripts, trans, by Child, p. 196.
112 A Century of Anticipation
what extent this is indebted to the earlier work of Kepler is difficult to
determine. Cavalieri emphatically denied any inspiration from Kepler's
method, other than "the names of a few solids and the admiration
which frequently sets philosophers to reflecting." 56 It is not improbable,
however, that the influence of Kepler upon Cavalieri may have resulted
indirectly from the correspondence of both of these men with Galileo. 67
A work upon indivisibles which Galileo planned to write never ap-
peared, but his views upon the subject are clearly brought out in his
classic treatise, the Two New Sciences, which was published three years
after the G'eometria of Cavalieri. Galileo's opinions resemble strongly
those expressed by his pupil, Cavalieri, and may well have been the
source of the latter's inspiration. 68 It may be well, therefore, to examine
at this point the views of Galileo, the teacher.
The forces molding the thought of Galileo and Cavalieri did not
differ greatly from those which had shaped the ideas of Kepler. These
men had mastered the Greek geometric methods, but they all betray
the effects of Scholastic speculations and the Platonic view of mathe-
matics which had exerted such a strong influence since the time of
Nicholas of Cusa. Both Galileo and Cavalieri were probably acquainted
with the modifications of the method of Archimedes by Valeric
Galileo referred to Valerio several times, in the Two New Sciences, as
the great geometer and as the new Archimedes of his age. 69 Galileo
gave in this work also an Archimedean demonstration of the quadra-
ture of the parabola 60 and included in an appendix some work on
centers of gravity in the manner of Commandino and Valerio. 61
Nevertheless both he and Cavalieri appear to be more significantly in-
debted, in attitude and method, to the later medieval speculations on
motion, indivisibles, the infinite, and the continuum. 62
The influence of Scholastic thought is clearly evidenced in the case
of Galileo by his early writings. In these he considered, among other
things, the Peripatetic doctrines of matter and form and of causes and
qualities, and the Scholastic questions of intension and remission and
of action and reaction. 63 On the latter he referred specifically to works
68 Exercitationes geometricae sex, pp. 237-38.
"Paul Tannery, Notions hisforiques, p. 341; Moritz Cantor, Vorlesungen, II, 774-75.
68 Paul Tannery, ibid., p. 341. 69 Le open di Galileo Galilei, VIII, 76, 184, 313.
•o Opere, Will, 181 ff . " Opere, VIII, 313 ; I, 187-208.
M Wallner, "Die Wandlungen des Indivisibiliensbegriffs von Cavalieri bis Wallis."
"Opere, I, "Iuvenalia," 111 ff., 119 ff., 126 ff.
A Century of Anticipation
113
by Calculator and Hentisbery. 64 In dynamics he made fundamental
the doctrine of impetus, which had been suggested by the Scholastic
philosopher Buridan in the fourteenth century and which was cer-
tainly familiar to Nicholas of Cusa, Leonardo da Vinci, and others of
the fifteenth and sixteenth centuries. 65 Whether Cusa's influence on
Galileo was significant (as it was on Kepler and probably also on
Giordano Bruno) is difficult to determine; 66 but that the Scholastic
discussions on motion were known to him will be very strongly indi-
cated by an examination of one of the propositions in his Two New
Sciences.
E
G
j
/
/
/
F
FIGURE 10
D
It will be recalled that Calculator and Hentisbery had demonstrated
dialectically that the average velocity of a body moving with uniform
acceleration is given by its velocity at the mid-point of the time in-
terval. Oresme had given a geometrical demonstration of this propo-
sition, in which he indicated that the area under the line representing
the velocity was the measure of the distance. Galileo's demonstration
of this proposition resembles strikingly that of Oresme. Let AB (fig. 10)
represent the time in which the space CD is traversed by a body which
84 IMA-, I, 172. •« Duhem, Etudes sur Leonard de Vinci, Vol. Ill, passim.
•• Goldbeck, "Galileis Atomistik und ihre Quellen."
114 A Century of Anticipation
starts from rest and is uniformly accelerated. Let the final speed be
represented by EB. Then the lines drawn parallel to EB will represent
the speeds of the body. It appears, then, that they may be interpreted
also as the moments, or infinitesimal increments, in the distance
covered by the moving body. Then the movements of the uniformly
accelerated motion may be represented by the parallels of triangle
AEB, whereas the parallels of the rectangle ABFG represent the cor-
responding moments of a body moving uniformly. But the sum of all
the parallels contained in the quadrilateral ABFG is equal to the sum
of those contained in the triangle A EB. Hence it is clear that the dis-
tances covered by the two bodies are equal, inasmuch as the triangle
and the rectangle are equal in area if / is the midpoint of FG. 67
Not only did Galileo reproduce with striking fidelity the diagram
and argument of Oresme as outlined above, but he extended a remark
made in this connection by Calculator and Hentisbery. Galileo could
have read in the works of these men, or in commentaries upon them,
that in uniformly accelerated motion the space covered in the second
half of the time is three times that covered in the first half. 68 This ob-
servation Galileo extended to show that if we subdivide the time in-
terval further, the distances covered, in each of these, will be in the
ratio 1, 3, 5, 7, ... 69 This is, of course, equivalent to the result ex-
pressed by the formula 5 = %gt 2 , and is implied by the earlier work of
the Scholastics.
The difference between Galileo's demonstration and that of Oresme
is largely one of completeness. Oresme had been satisfied to say merely
that inasmuch as the triangles EIF and AIG are equal, the distances
must be the same, thus implying the infinitesimal considerations neces-
sary to demonstrate that the areas represent distances. The geometrical
demonstrations of Oresme and Galileo are based upon the supposition
that the area under a velocity-time curve represents the distance
covered. 70 Since neither one possessed the limit concept, each resorted,
explicitly or implicitly, to infinitesimal considerations. Galileo ex-
pressed this view when he said that the moments, or small increments
in the distance, were represented by the lines of the triangle and the
*Opere, VIII, 208-9.
«» Duhem, Etudes sur Leonard de Vinci, III, 480, 513. 69 Opere, VIII, 210 ff.
70 Paul Tannery, "Notions historiques," pp. 338-39, mistakenly attributes this idea to
Galileo, rather than Oresme.
A Century of Anticipation 115
rectangle, and that these latter geometrical figures were in actuality-
made up of these lines. Galileo did not make clear how the transition
from the lines as velocities to the same lines as moments is to be made.
Oresme likewise had begged the question when he had represented in-
stantaneous velocities by lines and yet had maintained that all ve-
locities act through a time. Galileo and Oresme patently employed the
uncritical mathematical atomism which has appeared among mathe-
maticians of all ages — in Democritus, Plato, Nicholas of Cusa, Kepler,
and many others. It has been suggested that Galileo was influenced in
his use of indivisibles by his strong Pythagorean and Platonic approach
to science, 71 or by the revival in his day of interest in the atomism of
Heron of Alexandria, in which Galileo failed to distinguish clearly be-
tween physical indivisibles and mathematical infinitesimals. 72 He may
equally well have been led to his views by the Scholastic discussions of
infinitesimals.
At any rate, in the dialogue of the first day, in the Two New Sciences,
Galileo entered into an extended discussion of the subjects which had
been so popular in the medieval period: the infinite, the infinitesimal,
and the nature of the continuum. Galileo clearly admitted the possi-
bility of the Scholastic categorematic infinity, but because of the nu-
merous paradoxes to which this appeared to lead, Galileo concluded
that "infinity and indivisibility are in their very nature incompre-
hensible to us." 73 Nevertheless, he made at least one trenchant ob-
servation upon this subject. It will be recalled that Calculator had
remarked that there can be no ratio between an infinite magnitude and
a finite one. Galileo asserted more generally "that the attributes
'larger,' 'smaller,' and 'equal' have no place either in comparing in-
finite quantities with each other or in comparing infinite with finite
quantities." 74 In justifying this conclusion, Galileo indicated a sig-
nificant shift of emphasis, for instead of considering the infinite from
the point of view of magnitude, as had Aristotle and many medieval
scholars, he focused attention, as had Plato, upon the infinite as
multiplicity or aggregation. In this connection he indicated that the
"Wiener, "The Tradition behind Galileo's Methodology"; cf. also Brunschvicg, Les
lilapes de la philosophie mathematique, p. 70.
72 See Schmidt, "Heron von Alexandria im 17. Jahrhundert"; also other articles on
Heron in the same volume.
73 Opere, VIII, 76 ff . » Opere, VTH, 82 ff .
116 A Century of Anticipation
infinite class of all positive integers could be put into a one-to-one cor-
respondence with a part of this class — for example, with all the perfect
squares. 76 This characteristic of infinite sets was rediscovered in the
nineteenth century by Bolzano and later in that century was made
fundamental in the establishment of the calculus upon a rigorously
developed theory of infinite assemblages. 76
In spite of the trenchant observations which Galileo made on the
subject of the infinite, he felt strongly the inability of intuition to grasp
this notion. He went so far as to conjecture that there might be a third
possible type of aggregation between the finite and the infinite. This
seems to have been suggested to him by the difficulties of making pre-
cise our vague ideas on the continuum. He maintained, contrary to
Bradwardine, that continuous magnitudes are made up of indivisibles.
However, inasmuch as the number of parts is infinite, the aggregation
of these is not one resembling a very fine powder but rather a sort of
merging of parts into unity, as in the case of fluids. 77 This analogy is a
beautiful illustration of the effort which men made to picture in some
way the transition from the finite to the infinite.
Galileo sought also to clarify to some extent the paradoxes on mo-
tion. This he did by regarding rest as an infinite slowness, thus resort-
ing again to the vague feeling for the continuous to which Cusa and
Kepler had sought to give expression. In enlarging upon this idea,
Galileo applied to a falling body the argument which Zeno had given
in the dichotomy and then answered this by an appeal to intuition in
reversing the descent. In ascent, the body passes through an infinite
number of grades of slowness, finally to come to rest. The beginning
of motion is precisely the same, except that the order is reversed. 78
This argument constitutes a recognition — found also in Aristotle — of
the similarity of the difficulties in the dichotomy and the Achilles. It
represents, as well, an attempt to clarify the sense in which an infinite
series may be said to have a sum. Later Newton in his calculus, when
he spoke of his instantaneous velocities as prime or ultimate ratios,
made an analogous appeal to the fact that the beginning and the end
of motion are to be similarly conceived. Galileo appears not to have
realized, as Newton did somewhat vaguely, that only in terms of the
™Ibid.,Vni, 78.
76 Kasner, "Galileo and the Modern Concept of Infinity," pp. 499-501.
77 Opere, VTII, 76 ff . n Opere, VIII, 199-200.
A Century of Anticipation 117
limit concept can precise meaning be given to either the sum of an
infinite series or to a first or last ratio.
Whatever the influences which shaped the thought of Galileo, those
felt by his friend and student, Cavalieri, are not likely to have been
greatly different. Although familiar with the views of Valerio and
perhaps also with those of Stevin, 79 Cavalieri did not develop the limit
idea which these men had adumbrated. Instead he had resort to the
less subtle notion of the indivisible which had been adopted by Galileo. 80
However, whereas Galileo had employed this in physical explanation,
Cavalieri made it the basis of a geometrical method of demonstration
which achieved remarkable popularity. This method had been de-
veloped by Cavalieri as early as 1626, for in that year he wrote to
Galileo, saying that he was going to publish a book on the subject. 81
This work appeared in 1635, as the Geometria indivisibilibus continu-
orum nova quadam ratione promote,* 2 and the method it presented
was further developed a dozen years later in the Exercitationes geome-
tricae sex.
Cavalieri at no point in his books explained precisely what he under-
stood by the word indivisible, which he employed to characterize the
infinitesimal elements used in his method. He spoke of these in much
the same manner as had Galileo in referring to the parallel lines rep-
resenting velocities or moments as making up the triangle and the
quadrilateral. Cavalieri conceived of a surface as made up of an indefi-
nite number of equidistant parallel lines and of a solid as composed of
parallel equidistant planes, 83 these elements being designated the in-
divisibles of the surface and of the volume respectively. Although he
recognized that the number of these must be indefinitely great, he did
not follow his master Galileo in speculations as to the nature of the
infinite. The attitude of Cavalieri toward infinity was one of agnosti-
cism. 84 He did not share the Aristotelian view of infinity as indicating
a potentiality only — a conception which, in conjunction with the work
of Stevin and Valerio, pointed toward the method of limits. On the
other hand, Cavalieri did not join Nicholas of Cusa and Kepler in re-
79 Bosmans, "Les Demonstrations par l'analyse infinitesimale," p. 211.
80 Marie, Histoire des sciences mathematiques et physiques, III, 134.
a Moritz Cantor, Vorlesungen, II, 759.
82 1 have used the later edition, Bononiae, 1653.
85 Exercitationes geometricae sex, p. 3.
84 Cf. Brunschvicg, Les tiapes de la philosophic maihematique, p. 166.
118 A Century of Anticipation
garding the infinite as possessing a metaphysical significance. It was
employed by him solely as an auxiliary notion, comparable to the "so-
phistic" quantities of Cardan. Inasmuch as it did not appear in the
conclusion, its nature need not be made clear. That the infinite did
not enter explicitly into the arguments of Cavalieri was due to the fact
that at every stage he centered attention upon the correspondence
between the indivisibles of two configurations, rather than upon the
totality of indivisibles within a single area or volume. The proposition
still known in solid-geometry textbooks as Cavalieri' s Theorem is char-
acteristic of his approach: If two solids have equal altitudes, and if
sections made by planes parallel to the bases and at equal distances
from them are always in a given ratio, then the volumes of the solids
are also in this ratio. 86
FIGURE 11
Typical of the propositions in the method of indivisibles, and of far-
reaching significance in later developments, were a number of theorems
on the lines of a parallelogram and those of its constituent triangles.
One of these propositions consisted in an extended demonstration that
if a parallelogram AD (fig. 11) is divided by the diagonal CF into two
triangles, ACF and DCF, then the parallelogram is double either
triangle. 86 This is proved by showing that if one lay off EF = CB and
draw HE and BM parallel to CD, then the lines HE and BM are equal.
Therefore all the lines of triangle ACF taken together are equal to all
those of CFD; consequently triangles ACF and CDF are equal, and
w Geometria indivisibilibus, pp. 113-15; Exercitationes geometricae sex, pp. 4-5. Cf. also
Evans, "Cavalieri's Theorem in His Own Words."
M Geotnetria indivisibilibus, Proposition XIX, pp. 146-47; Exercitationes geometricae sex,
Proposition XIX, pp. 35-36.
A Century of Anticipation 119
the sum of the lines of the parallelogram AD is double the sum of the
lines of either triangle.
From here Cavalieri went on to prove by a similar, but considerably
more involved, argument that the sum of the squares of the lines in the
parallelogram is three times the sum of the squares of the lines in each
of the constituent triangles. 87 Using this latter proposition, he then
easily demonstrated, among other things, that the volume of a cone is
§ that of the circumscribed cylinder, and that the area of a parabolic
segment is § the area of the circumscribed rectangle. 88 These results
were of course known to Archimedes, but a problem which Kepler had
proposed some years before now led Cavalieri to a use of indivisibles
bolder than that which is found in the Geometric, and to a new result
of significance. 89
Kepler had, in his Stereometria, challenged geometers to find the
volume of the solid obtained by rotating a segment of a parabola about
its chord. 90 Cavalieri determined this volume by basing the problem
on the discovery that the sum of the fourth powers of the lines of a
parallelogram is 5 times the sum of the fourth powers of the lines of
one of the constituent triangles. Then he recalled that in his Geometria
indivisibilibus he had found the ratios of the lines to be 2 to 1 and the
ratio of the squares of the lines to be 3 to 1. In order not to leave a gap
in his results on the ratios of the powers of the lines of a parallelogram
and of the triangle, Cavalieri sought the ratio for the sums of the cubes
of the lines, and found this to be as 4 to 1. He then concluded by anal-
ogy that for the fifth powers it would be 6 to 1, for sixth powers 7 to
1, and so on, the sum of the nth powers of the lines of the parallelogram
being to the sum of the nth powers of the lines of the triangle as n +
1 to l. M
The method of Cavalieri here employed was based upon several
lemmas which are equivalent to special cases of the binomial theorem. 92
For example, to prove that the sum of the cubes of the lines of a paral-
87 Geometria indivisibilibus, Proposition XXIV, pp. 159-60; Exercitationes geometricae
sex, Proposition XXIV, pp. 50-51.
88 Geometria indivisibilibus, pp. 185, 285-86; Exercitationes geometricae sex, pp. 78 ff.
89 For a summary of this work, see Bosmans, "Un Chapitre de l'oeuvre de Cavalieri."
80 Opera omnia, IV, 601.
81 Exercitationes geometricae sex, pp. 290-91. For a statement of the manner in which he
was led to this result, see pp. 243-44.
92 Exercitationes geometricae sex, p. 267.
120 A Century of Anticipation
lelogram is four times the sum of the cubes of the lines of one of the
constituent triangles, he began with (a -f b) 3 = a 3 + 3a 2 b + 3ab 2 -f b 3 .
Then he proceeded somewhat as follows: setting AF = c, GH = a,
HE = b, in figure 11, we have 2c 3 = 2a 3 + 32a 2 6 + 32ab 2 + 2& 3 ,
where the sums are taken over the lines of the parallelogram and tri-
angles. By symmetry this can be written 2c 3 = 22a 3 + 62a 2 6. Now
2c 3 = cZc 2 - c2 (a + b) 2 = cZa? + ICLab + c2& 2 . But in an earlier
proposition on lines Cavalieri had shown that 2a 2 = 2& 2 = \l>c 2 . This
gives us 2c 3 = |c2c 2 ■+ 2c2a&
= §2c 3 + 2 (a + b)2ab = |2c 3 + 22a 2 6 + 22& 2 a
= 12c 3 + 42a 2 6
or 2a 2 6 = tV2c 3 . Substituting this in the equation above, we obtain
2c 3 = 22a 3 + ^2c 3 , or 2c 3 = 42a 3 , and the proposition is proved. 93
Cavalieri realized that this method can be generalized for all values
of «, 94 but he gave complete demonstrations only up to and including
n — 4. For higher values he gave only some "cossic" indications, which
had been given him by Beaugrand. 95 These may well have been derived
from the contemporaneous work of Fermat, 96 whose methods we shall
consider later.
The results of Cavalieri outlined above are in a general sense equiva-
a n + 1
lent to what we would now express by the notation J x dx = .
n + 1
We have seen that Archimedes, through the use of series in connection
with his stereometric work, had recognized the truth of this proposi-
tion for n = 1 and n = 2. He may have known of it for n = 3 also, and
the Arabs proved it for n = 4 as well. 97 Cavalieri's work, although
based upon somewhat different views, was a generalization of these
results of Archimedes and the Arabs. It will be seen later that in this
respect Cavalieri had been anticipated by several contemporary
mathematicians. Nevertheless, his statement, which appeared in 1639, 98
represents the first publication of this theorem, which played a signifi-
» Ibid., pp. 273-74. M "Et sic in infinitum." Ibid., p. 268.
95 Ibid., pp. 286-89. 9 « See Fermat, CEuvres, Supplement, p. 144.
87 See Paul Tannery, "Sur le sommation des cubes entiers dans l'antiquitfi," and Ibn
al-Haitham (Suter), "Die Abhandlung uber die Ausmessung des Paraboloids."
88 Cenluria di varii problemi, pp. 523-26. Simon ("Zur Geschichte und Philosophic der
Differentialrechnung," p. 118) has said — very probably through some error — that Cava-
lieri found this result in 1615.
A Century of Anticipation 121
cant role in the development of infinitesimal methods during the period
from 1636 to 1655. Within this interval, the mathematicians Torricelli,
Roberval, Pascal, Fermat, and Wallis arrived, all more or less inde-
pendently and by varying methods, at this fundamental result and
extended it, as well, to include negative, rational fractional, and even
irrational values of n. It was perhaps the first theorem in infinitesimal
analysis to point toward the possibility of a more general algebraic
rule of procedure, such as that which, formulated a generation later
by Newton and Leibniz, became basic in the integral calculus. Cava-
lieri himself had no vision of such a new analysis; neither he nor Galileo
appears to have been seriously interested in algebra, either as a manner
of expression or as a form of demonstration. The proposition remained
for Cavalieri a geometrical theorem concerning the ratio subsisting be-
tween a sum of powers of the lines of a parallelogram and that of one
of the constituent triangles. Furthermore, he did not express any clear
conception — such as is basic in our idea of the definite integral — of
his ratios as limits of sums of infinitesimal parallelograms. Cavalieri
never did make clear his interpretation of the indivisible and in this
respect laid his method open to attack.
Cavalieri's use of indivisibles in his Geometria had been criticized by
the Jesuit, Paul Guldin, who asserted not only that the method had
been taken from Kepler, but also that it was incorrect, inasmuch as it
led to paradoxes and fallacies. Cavalieri defended himself from the
first charge by pointing out that his method differed from that of
Kepler in that it made use only of indivisibles, whereas Kepler had
thought of a solid as made up of very tiny solids." In answering the
charge that the method was invalid, Cavalieri maintained that al-
though the indivisibles may correctly be considered as having no
thickness, nevertheless, if one wished, he could substitute for them
small elements of area and volume in the manner of Archimedes.
Guldin had said that since the number of indivisibles was infinite,
these could not be compared with one another. Furthermore, he had
w "Ex minutissimis corporibus." Exercitationes geometricae sex, p. 181; see also Kepler,
Opera omnia, IV, 656-57, for notes, including extracts from Cavalieri on this point.
Cavalieri's assertion in this connection should be sufficient to refute the statement often
made (see e. g., Wolf, A History of Science, Technology, and Philosophy in the Sixteenth and
Seventeenth Centuries, p, 206) that he was probably well aware of the fact that his indi-
visibles must be of the same dimension as that of the figure which they constitute.
122 A Century of Anticipation
pointed out a number of fallacies to which the method of indivisibles
appeared to lead. In answering these arguments, Cavalieri said that
difficulties in the method are avoided by observing that the two
infinities of elements to be compared are of the same kind. If, for
example, the altitudes of two figures are unequal, their horizontal sec-
tions are not to be compared, because the corresponding indivisibles
of one are not the same distance apart as are those in the other. Where-
as in one figure there may be 100 indivisibles between two sections,
between the corresponding sections of the other there may be 200. ,0 °
This explanation Cavalieri followed with an ingenuous comparison
of the indivisibles of a surface with the threads of a piece of cloth, and
those of a solid with the pages of a book. 101 Although in geometrical
solids and surfaces the indivisibles are infinite in number and lacking
in all thickness, nevertheless they may be compared in the same manner
as in case of the cloth and the book, if one observes the precaution
mentioned above. Cavalieri did not explain how an aggregate of ele-
ments without thickness could make up an area or volume, although
in a number of places he linked his method of indivisibles with ideas
of motion. This association had been suggested somewhat vaguely by
Plato and the Scholastic philosophers, and Galileo had followed them
in associating dynamics with geometrical representation. Napier, in
1614, had likewise employed the idea of the fluxion of a quantity to
picture by means of lines the relation between logarithms and num-
bers. 102 Cavalieri followed this trend in holding that surfaces and
volumes could be regarded as generated by the flowing of indivisibles. 103
He did not, however, develop this suggestive idea into a geometrical
method. This was done by his successor Torricelli, with the result that
it eventuated later in Newton's method of fluxions. Cavalieri's indi-
visible itself was also to find a counterpart in the thought of both
Newton and Leibniz — in the former's conception of moments and in
the latter's notion of differentials. Cavalieri's vague suggestions were
thus to play a large part in the development of the calculus.
The Geometria indivisibilibus of Cavalieri achieved popularity al-
most immediately, and became, except for the works of Archimedes,
100 Exercitationes geometricae sex, pp. '238-39; cf. also p. 17.
101 Ibid., pp. 239-40; cf. also pp. 3-4.
m Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert, pp. 134-35.
108 Exercitationes geometricae sex, pp. 6-7; Geometria indivisibilibus, p. 104.
A Century of Anticipation 123
the most quoted source for mathematicians dealing with infinitesimal
considerations in geometry. The significance of the work in the history
of mathematics has been justly recognized, but such recognition has
on occasion led to attempts to impute to Cavalieri views which he was
far from possessing. It has been asserted that in his work one discerns
clearly "the fundamental notion of the differential calculus," inasmuch
as "the indivisible is nothing but the differential," 104 and that one dis-
covers in it "definite integrals in the sense of Cauchy and Riemann." 108
An examination of his ideas and methods will show that such judg-
ments are not sound. Cavalieri was far from possessing the views which
are expressed in the terms "differential" and "integral." He himself
appears to have regarded his method only as a pragmatic geometrical
device for avoiding the method of exhaustion; the logical basis of this
procedure did not interest him. Rigor, he said, was the affair of phi-
losophy rather than geometry. 106 The limit idea, toward which Stevin
and Valerio were working, was more completely concealed in Cava-
lieri's method than in Kepler's. Furthermore, there is in the Geometria
indivisibilibus a complete lack of emphasis on the algebraic and
arithmetical elements which were to lead, first to the rules of procedure
of the calculus and later to the satisfactory definitions of the differ-
ential and the integral. Cavalieri regarded area and volume as in-
tuitively clear geometrical concepts, and invariably determined the
ratio of these, rather than a numerical value associated with a single
area or volume. This preoccupation with ratios was to be one of the
chief causes of the confusion in the basic ideas of the calculus during
the following two centuries.
The fact that Cavalieri paid so little heed to the demands of mathe-
matical rigor made geometers chary of accepting the method of in-
divisibles as valid in demonstrations, although they employed it
readily in preliminary investigations. This element of hesitation is seen
to particularly good advantage in the work of Evangelista Torricelli,
the friend of Cavalieri and the pupil of Galileo.
Torricelli fully realized the advantages and disadvantages of the
method of indivisibles; and he suspected that the ancients possessed
104 Milhaud, "Note sur les origines du calcul infinitesimal," pp. 37-38.
106 Bortolotti, "La scoperta e le successive generalizzazioni di un teorema fondamentale
di calcolo integrale," p. 210, n.
108 Exercitationes geometricae sex, p. 241.
124 A Century of Anticipation
some such method for discovering difficult theorems, the proofs of
which they cast in another form either "to hide the secret of their
method or to avoid giving occasion for contradiction to jealous de-
tractors." 107 The work of Archimedes indicates how correct Torricelli
was in his assumption of the existence of this method, but the motive
behind the proofs by exhaustion lay rather in an effort to satisfy the
Greek demands for intuitive clarity and logical rigor. Torricelli himself
was not quite satisfied with demonstrations by the method of indi-
visibles, for he usually supplemented these by proofs in the manner of
Archimedes, or of Valerio, to whom he refers (echoing Galileo) as the
"Archimedes of our century." 108
The De dimensions parabolae of Torricelli is an interesting exercise,
in which the author offers twenty-one demonstrations of the quadra-
ture of the parabola. In ten of these the proposition is established by
the method of the ancients, 109 and in the other eleven by the new
geometry of indivisibles. 110 Included among the former 111 is the well-
known demonstration by the method of exhaustion, given by Archi-
medes in his Quadrature of the Parabola. There is, as well, a proof 112
closely resembling the basic proposition of Luca Valerio on inscribed
and circumscribed figures; viz., that it is possible to inscribe within
the parabolic segment a figure, made up of parallelograms of equal
height, which shall differ from the segment by less than any given
magnitude*. Torricelli did not, of course, consider the area of the parab-
ola as defined by the limit of the inscribed figure, but he approached
more closely to this idea than had Valerio. The latter had been satisfied
to state that the circumscribed and inscribed figures differed by less
than a given magnitude and to imply that the figure itself would there-
fore differ from either of these by less than this magnitude; Torricelli
clearly stated this implication, from which the limit concept would
easily follow on arithmetizing the quantities involved. However, Torri-
celli followed Cavalieri in restricting himself to geometrical considera-
tions, and in consequence he was attracted more toward indivisibles
than toward limits.
107 Opere di Evangdista Torricelli, I (Part 1), 140. »* Opere, I (Part 1), 95.
io» "More antiquorum absolute." Opere, I (Part 1), 102.
uo »p er novam indivisibilium geometriam pluribus modis absoluta." Opere, I (Part 1),
139.
111 Proposition V. Opere, I (Part 1), 120-21. 1U Lemma XV. Opere, I (Part 1), 128-29.
A Century of Anticipation
125
Among the eleven demonstrations by the geometry of indivisibles,
he included one 113 which is— oddly enough— almost identical with the
mechanical quadrature given by Archimedes in the Method, a work not
known in the seventeenth century. This coincidence shows how closely
Cavalieri's geometry of indivisibles resembled the mathematical
atomism upon which Archimedes' method was probably based.
Torricelli far outdid his master Cavalieri in the flexibility and per-
spicuity of his use of the method of indivisibles in making new dis-
coveries. One of the novel results which pleased him greatly was the
M
FIGURE 12
determination, in 1641, 114 that the volume of an infinitely long solid,
obtained by revolving about its asymptote a portion of the equi-
lateral hyperbola, was finite. 116 Torricelli believed that he was the first
to discover that a figure with infinite dimensions could have a finite
magnitude; but in this respect he had been anticipated, probably by
Fermat and Roberval, and certainly in the fourteenth century by
Oresme. It will be recalled that Oresme, in applying geometric repre-
113 Proposition XX. Opere, I (Part 1), 160-61.
114 Bortolottj, "La memoria 'De infinitis hyperbolis' di Torricelli," p. 49.
U8 "De solido hyperbolico acuto." Opere, I (Part 1), 173-221.
126 A Century of Anticipation
sentations to the considerations of Calculator on the latitude of forms
and infinite series, had shown that a figure with an infinite altitude
(velocity) could nevertheless have a finite area (distance). Torricelli's
proof is interesting in that it makes use of the idea of cylindrical in-
divisibles, whereas those of Cavalieri had invariably been plane. Let
the hyperbola be rotated about BA, and let ED be a fixed horizontal
line (fig. 12). Let ACGH be a right circular cylinder with AC as alti-
tude and A H as the diameter of its base.* Then Torricelli showed that
for any position of the line NL parallel to ED, the cylinder with alti-
tude NO and diameter 01 has a lateral surface area equal to the cross-
sectional area IM of the cylinder ACGH. But the cylindrical surfaces
NLIO make up the volume of the infinitely long solid of revolution
FEBDC; and similarly the areas of the circles of diameter IM con-
stitute the volume of the cylinder ACGH. Therefore the two volumes
are equal. 116
The demonstration given by Torricelli shows strikingly the facility
offered by the comparison of indivisibles, and the extent to which it
resembles the procedure employed in the integral calculus, in which,
of course, the cylindrical and the circular elements are given a thick-
ness, and the limit of the sum, as this thickness approaches zero, is de-
termined. Torricelli said that he himself was satisfied that the truth
of this theorem was sufficiently clear; but for the benefit of those not
so kindly disposed toward indivisibles, 117 he added, as usual, a demon-
stration by the method of the ancients. 118
The work of Torricelli is so eminently catholic in its application of
the ideas and methods suggested by his predecessors and contempo-
raries that his name on numerous occasions became the center of dis-
putes concerning priority. Probably no century presents more charges
of plagiarism than does the seventeenth. This is accounted for largely
by the fact that the method of indivisibles and related procedures were
used extensively and effectively by mathematicians of many nation-
alities, all working on similar problems which were leading toward the
calculus. Inasmuch as there was no logically established justification
for these heuristic infinitesimal methods, the interrelations of the di-
ne "Propterea omnes simul superficies cylindricae, hoc est ipsum solidum acutum EBD,
una cum cylindro basis FEDC, aequale erit omnibus circulis simul, hoc est cylindro
ACGH." Opere, I (Part 1), 194.
117 Ibid. u« Und, t pp . 214-21.
*Where AH is twice the distance from A to the hyperbola.
A Century of Anticipation 127
verse points of view were but vaguely realized, and were frequently
denied. Added to this was the further difficulty that many contributors
to the theory of the subject— notably Roberval, Fermat, and Newton
—either did not publish their results, or did so only very tardily.
Failure of mathematicians to date their works further added to the
confusion experienced in attempting to attribute specific contributions
to individual men. No attempt will be made in this essay to consider
at length such charges of plagiarism, many of which were not sufficiently
substantiated. Nevertheless, an effort will be made to make clear, so
far as is possible in this difficult period, the part each man played in
the development of the ideas of the calculus.
It will be recalled that Cavalieri had enunciated in geometrical ter-
minology what may be considered the first general theorem of the
n + 1
calculus: that is, the equivalent of j a Q x n dx = — — -, for all positive in-
n -f 1
tegral values of n. The generalization and proof of this theorem for all
rational values of n (except n = — 1) has been commonly attributed to
Fermat. 119 However, inasmuch as Fermat did not publish his result
during his lifetime, it is difficult to determine the relationship between
his work and the analogous and approximately concurrent results of
Torricelli and Roberval. It would appear 120 that the enunciation, at
least, of the generalization is to be attributed to Fermat. The date of
this is doubtful. It may have been as early as 1635, 121 or as late as
1643. 122 Whether the demonstrations of the rule given by Roberval for
positive integers and by Fermat for the general rational case antici-
pated that given by Torricelli in 1646, in his unorganized treatise, De
infinitis hyperbolis, is not clear. 123 However, the form of proof employed
119 See, for example, Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert,
p. 265.
120 Bortolotti, "La scoperta," p. 215.
ia See Walker, A Study of the TraitS des Indivisibles of Roberval, pp. 142-64.
» Bortolotti, "La Scoperta," p. 215.
m Zeuthen asserts that it is to Fermat that we owe the study of the higher parabolas.
He says that Fermat was probably in possession of the rule for positive integers in 1636
and that it is almost certain that he had a general demonstration for all cases in 1644.
"Notes sur l'histoire des mathematiques," IV. "Sur les quadratures avant le calcul inte-
gral, et en particulier sur celles de Fermat," (1895), pp. 43 ff. Surico, on the other hand,
maintains the priority of Torricelli in this connection, placing the discovery in 1641 and
the generalization by 1646. He concludes that Fermat's quadratures were posterior to
1654, perhaps about 1656. "L'integrazione di y = x n per n negativo, razionale, diverso
da — 1 di Evangelista Torricelli."
128 A Century of Anticipation
is in each case so peculiar to the ideas of its author that, in considering
the development of the concepts of the calculus, we may, with-
out fear of misdirection, consider them as independent of each
other.
The demonstration offered by Torricelli is in the manner of Archi-
medes — purely geometrical and employing the method of exhaustion.
Given any hyperbola DC (fig. 13), he proved that "the quadrilineum
EDCF is to the frustum DCBG as the power (dignitas) of BA is to the
power of AE." m The proof of this he carried out by the use of in-
scribed and circumscribed figures and the application of several
lemmas, including the fundamental proposition of Valerio, in which
it is shown that these figures can be made to differ by less than any
given area. Torricelli remarked, incidentally, that the same procedure
and conclusion will also be found to apply, with slight alterations, in
the case of parabolas.
This result is equivalent to the analytic statement that if the curve
is x m y n = k, then the ratio of the areas EDCF and DCBG is — . The
m
determination of this ratio is in a general sense equivalent to the
/b m
a x~ n dx. In such a rep-
resentation of the problem in terms of modern symbolism, 125 however,
there is a strong temptation to read into the author's work the con-
cepts which are called to mind by the newer notation. One must not
attribute to Torricelli any such algebraic notion as that implied by the
modern integral sign. Analytic considerations at no point enter into
his thought, and there is no indication that he had any desire to es-
tablish an algorithmic rule of procedure applicable to other cases. The
result remained a simple geometrical proposition on the ratio of areas,
although the particular quadrature involved was later to be of funda-
mental significance in the calculus.
The danger in interpreting Torricelli 's work in terms of modern no-
tations and ideas is apparent in an evaluation of his work on tangents.
Torricelli discovered that if DT is tangent at D to the hyperbola con-
sidered above (fig. 13), then TE is to EA as the ratio of the powers of
» Opere, I (Part 2), 256.
125 Such as that given at some length by Bortolotti, "La memoria."
A Century of Anticipation
129
AB and AE; m that is, if the hyperbola is x m y n = k, the ratio of the
ft • *
subtangent to the abscissa is — . An interpretation of this proposition
m
has been made, 127 in which it is implied that Torricelli regarded the
tangent as determined by the secant through C and D as C approaches
D, and that he was therefore not far from the idea of a differential
quotient. However, there is nothing in Torricelli's language to justify
such a conclusion. Torricelli's proof is based, not on the modern idea
of the tangent as the limit of a variable secant, but upon the ancient
E
FIGURE 13
static definition: a line touching the curve at only one point. He
showed that the assumption that the line TD, determined by the ratio
stated above, is not tangent at D— that is, that it intersects the curve
in another point — leads to a contradiction. There is in this proof, of
course, nothing of the modern idea of limits. 128 However, in another
connection Torricelli made use of a dynamic conception of tangents
which was significantly suggestive in the development of Newton's
fluxional calculus— that based on the parallelogram of virtual velocities.
™Opere, I (Part 2), 257.
m Bortolotti, "La memoria," pp. 143-44.
«• Cf. also Opere, I (Part 2), 304 ff.
130 A Century of Anticipation
The principle of the parallelogram of velocities may be considered
to be implied 129 in Peripatetic science, but inasmuch as Aristotle did
not develop the notion of instantaneous velocity, the doctrine long
failed to be widely employed. Archimedes appears to have applied it
in geometry, and much later it was again suggested by Leonardo da
Vinci, 130 and used by Stevin, 131 who, because of a predominant interest
in statics, thought of it in terms of virtual displacements rather than
velocities. Upon the clarification by Galileo of trajectories in terms of
the notion of inertia and of the doctrine of the independence of super-
imposed effects, the idea of the composition of motions was destined
to play a significant role in science (particularly in dynamics and
optics) and mathematics during the seventeenth century.
Torricelli's determination, by means of the composition of motions,
of the tangents to parabolas of any positive integral degree furnishes
a striking illustration of his application of the methods of Galileo and
Cavalieri. It had been recognized since the fourteenth century that
the motion of a freely falling body is uniformly accelerated — that is,
that the velocity increases in proportion to the time elapsed — and
Galileo had incorporated this fact in his dynamics. It had been stated
by Calculator, Oresme, and others that the distance covered was con-
sequently one-half that which would be traversed by a body moving
uniformly for the same length of time with half of the final velocity of
the falling body. Galileo reasserted this and added that this implied
that the distance covered by the falling body varied as the square of
the time. Torricelli pursued this idea further and inquired what would
be the state of affairs if the velocity were to vary as the square of the
time. In this case the distance covered would be given by the sums of
the squares of the lines in the triangle ABE (fig. 14), where these
squares represent the velocities of the body for a given time interval
AB. But Cavalieri had demonstrated that the sum of the squares of
the lines of the triangle ABE is \ the sum of the squares in the par-
allelogram ABEL Therefore the distance covered will be \ that which
would have been covered by a second body moving for the time AB
with a uniform velocity equal to the final velocity of the first body —
129 See Duhem, Les Origines de la statique, II, 245.
130 Duhem, op. cit., II, 245-65, 347-48; cf. also Duhring, Kriliscke Geschichte der allge-
meinen Principien der Mechanik, p. 15.
m Lasswitz, Geschkltte der Atomistik, II, 12-13.
A Century of Anticipation
131
or, inversely, the final velocity will be given by three times the dis-
tance covered. The distance, moreover, will vary as the cube of the
time. 132
If, therefore, we imagine a projectile moving with a composite mo-
tion made up of a uniform horizontal velocity and of a vertical ve-
locity which varies as the square of the time, the curve traversed will
be a cubical parabola; if the vertical velocity varies as the cube of the
time, the curve will be a quartic parabola; and so on. The tangents to
H
FIGURE 14
these curves many now be determined as follows: Let the curve ABC
(fig. 15) be, for example, the cubic parabola. Then if EB is tangent to
the curve, we will have ED = 3 AD. This is clear from the fact that
the moving point will possess at B a double impetus: one horizontal,
and given by the distance BD: and the other vertical, and given (from
the considerations above) by three times the vertical distance AD.
Therefore the direction of the point B, by the composition of these two
velocities, will be that of the line BE, which is consequently the tan-
m Opere, I (Part 2), 311.
132
A Century of Anticipation
gent. 133 Torricelli remarked that the same type of argument can like-
wise be applied to other parabolas, the ratio - — being the degree of
JUJ
the parabola.
Torricelli's method, employing as it does the idea of instantaneous
direction and implying, therefore, the limit concept, represents a
marked advance over the stultifying definition of the tangent given
FIGURE 15
by the ancient geometers. It indicates, as well, a departure from the
classical tradition, in the intrusion of the notion of instantaneous ve-
locity into geometrical demonstrations. Dynamical considerations had
occasionally forced themselves upon the attention of Greek geometers.
Archimedes may well have used Torricelli's very method in determin-
ing the tangent to his spiral; but he did not, any more than his con-
»/wa., pp. 310-11.
A Century of Anticipation 133
temporaries, regard the idea of motion as sufficiently rigorous for
application in his static and formal geometrical proofs, nor did he even
develop a science of kinematics. The Scholastic philosophers of the
fourteenth century, however, had given quantitative representations
of dynamics, work which was elaborated by Galileo. Torricelli now
employed these in pure geometry in the determination of tangents to
parabolas.
He made use of them also in finding the tangents to a large class of
curves suggested to him by the work of Archimedes. Torricelli con-
sidered the curves generated by a point which moves along a uniformly
rotating line with a velocity which is not necessarily uniform, but is,
instead, a function of its distance from the fixed point about which the
line rotates. 134 If the velocity is such that in equal intervals of time the
distances of the variable point from the fixed point are in continued
proportion, Torricelli called the curve a "geometric spiral," to dis-
tinguish it from the "arithmetic spiral" of Archimedes. 135 The equation
of the geometric spiral may be written in polar coordinates as P =
ae hd , but Torricelli did not treat these curves analytically. Instead, he
employed considerations from synthetic geometry and mechanics to
give him the tangents, as well as the lengths of the curves and the
areas bounded by them.
In his use of kinematic representations in mathematics, he may have
been anticipated by the French mathematicians Roberval 138 and
Descartes (whom we shall consider shortly) ; but it was largely through
Torricelli's work that the notion gained such popularity that it was
accepted by Barrow in his geometry and by the latter 's student, New-
ton, in the method of fluxions.
Among the results achieved by Torricelli through the application of
the methods of exhaustion, of indivisibles, and of the composition of
motions are to be found a number of remarkable anticipations of those
found in the calculus. They include, as well as numerous theorems on
quadratures and tangents, some of the earliest results on rectification.
Torricelli appears also to have recognized and made use of the fact
that the problem of tangents was the inverse of that of quadratures. 187
M "De infinitis spiralibus," Opere, I (Part 2), 349-99. M Ibid., p. 361.
m See Jacoli, "Evangelista Torricelli ed il metodo delle tangenti detto Meiodo del
Roberval."
w Bortolotti, "La memoria," pp. 150-52.
134 A Century of Anticipation
He did not, however, attempt to establish upon these methods any
general rules of procedure which might be applied in all cases. He did
not regard them as constituting a new type of analysis and conse-
quently did not seek for a universal algorithm. The composition of
velocities, for example, could be employed only in the case of curves
for which the generating motions were known beforehand. Only with
the analytic methods of Fermat, Descartes, and Barrow, or with the
calculus of Newton and Leibniz did it become possible to determine
in general, from the equation of a curve, the motions by which the
curve could be regarded as traced, or the instantaneous direction.
Although Torricelli's work marked a significant step toward the
calculus, the basic concepts employed in it were still far from the
modern point of view. Torricelli had no more idea of defining the no-
tion of instantaneous velocity in terms of limits than had his master,
Galileo — or, for that matter, than his successor, Newton. Similarly,
although Torricelli was well aware of the fact that the method of in-
divisibles gave results consonant with those obtained by the methods
of the ancients, he appears to have been far from realizing that the
two are to be associated through the limit concept. His view of the
indivisible resembled strongly the vague mathematical atomism of
Democritus. Torricelli agreed with the assertion, which he attributed
to Galileo, 138 that a point is equal to a line; and in "confirming"
Galileo's demonstration that the area under a velocity-time graph
represents distance, Torricelli affirmed that in the case of unequal lines
the number of points on each was the same, but that the points them-
selves were unequal. 139 The lack of a suitable basis for indivisibles was
perhaps more serious than the omission of a definition of instantaneous
velocity, for whereas intuition may serve to clarify the use of velocity
as an undefined element, there is no such safe guide in the use of in-
divisibles. Torricelli realized the difficulties involved; but, although he
wrote a work on paradoxes in the use of infinitesimals, 140 he was unable
to resolve the logical perplexities.
The date 1647 is significant in the development of the calculus for
a number of reasons : in the first place Cavalieri and Torricelli died in
this year, both at early ages; secondly, Cavalieri's more ambitious
»* Opart, I (Part 2), 321.
"• Ibid., p. 259. uo De indivisibilvum doctrina per per am usurpata. Ibid., pp. 415-23.
A Century of Anticipation 135
work on indivisibles — the Excrcitationes geometricae sex — appeared;
thirdly, the ponderous Opus geometricum of the greatest of circle-
squarers, Gregory of St. Vincent, was published at Antwerp.
Let us now turn from the development of infinitesimals in Italy, to
consider briefly a concurrent trend in the Low Countries, of which the
point of view was somewhat different. Gregory of St. Vincent and
Cavalieri probably worked independently of each other 141 and were in
possession of their methods at practically the same time — about 1625
and 1626 respectively. 142 Both men were directly connected, in their
mathematical inspiration, with the tradition of Archimedes, and each
was directly indebted, as well, to others — Cavalieri to Galileo (from
whom he probably adopted his indivisibles), Valerio, and perhaps
Kepler; Gregory to Stevin and Valerio, from whom he borrowed the
idea of giving to propositions involving the method of exhaustion a
direct and yet rigorous demonstration in place of the Greek reductio
ad absurdum. 143 Gregory, however, added an element not found in
their works, for he connected the question with the Scholastic dis-
cussions on the nature of the continuum and the result of infinite
division. Archimedes, Stevin, and Valerio had subdivided only until
the error was less than a certain amount, but Gregory interpreted this
as meaning an actually infinite subdivision. Instead of the parallelo-
grams of Stevin and Valerio, he used, in his Opus geometricum (or
Problemum austriacum, as it is sometimes called), infinitely many
infinitely thin rectangles; 144 and for the w-sided polygons used by
Archimedes he substituted an inscribed polygon of an infinite number
of sides, 145 as had Nicholas of Cusa.
Gregory applied his conceptions to problems of cubatures by a
process which he called ductus plant in planum. This phrase referred
to a means of constructing a geometrical solid from two given plane
figures. Let the figures be, for example, a semicircle and a rectangle
having the diameter of the circle as one side (fig. 16). Then, by "ap-
plying the rectangle to the semicircle," Gregory had in mind the fol-
lowing: Place the rectangle ABA'B' perpendicular to the plane of the
semicircle. Then for any point on AB erect in the two figures perpen-
141 Moritz Cantor, Vorlesungen, II, 818.
ia Ibid., II, 759; see also Bosnians, "Gregoire de Saint- Vincent."
la Bosnians, op. til., pp. 250-56. "* Opus geometricum, p. 961.
144 Wallner, "Ober die Entstehung des Grenzbegriffes."
136
A Century of Anticipation
dicular lines XX' and XZ and upon these complete the rectangle
XZX'Z' . The geometrical solid, of which this rectangle is a section for
all positions of X on AB, is the one sought. In this case it is a portion
of a cylinder, but Gregory applied the process to innumerable other
figures 146 and found the volumes of the solids obtained.
It is obvious that in this procedure Gregory of St. Vincent was im-
plicitly making use of infinitesimals. His view of the nature of these,
although perhaps less naive than that of Cavalieri, was not clear or
rigorous. 147 The manner in which he constructed his solid figures would
suggest that he was thinking in terms of indivisibles. Nevertheless, we
FIGURE 16
see in other connections that he did not regard them, as had Cavalieri'
as being without thickness, but rather in the manner of Kepler, as
literally making up the geometrical figure. After inscribing in two three-
dimensional figures very thin parallelepipeds, he added that "these
parallelepipeds can be so multiplied that they exhaust the body within
which they are inscribed." 148
This is perhaps the first use of the word "exhaust" in this sense and
the earliest example in which a configuration was literally exhausted
in this manner. In Greek demonstrations by the method of exhaustion
146 Opus geometricum, Book VII. MT Moritz Cantor, Vorlesungen, II, 818.
148 "Parallelepipeda ilia ita posse multiplicari ut corpora ipsa, quibus inscribuntur,
exhauriant." Opus geometricum, p. 739.
A Century of Anticipation 137
the figure was thought of simply as approximated, to within a given
degree of accuracy, by the inscribed or circumscribed figure. The
Greek method is therefore, in a sense, improperly designated as the
method of exhaustion; but Gregory apparently used the word in its
literal sense, allowing the subdivision to continue to infinity. He did
not explicitly state how one is to visualize the exhaustion of the body
by means of infinitesimal elements, but he was certainly nearer to the
modern view than Democritus and the Scholastics, or even Cavalieri
and Torricelli, had been. Instead of thinking of static indivisibles, he
reasoned in terms of a varying subdivision, thus approximating the
method of limits. This fact led Gregory in the direction which was
ultimately to supply the rigorous basis for the calculus, for his infinite
subdivision brought him to the notion of the limit of an infinite geo-
metrical progression. 149
The Greeks had not attempted to define a curve as the terminus ad
quern, or limit, approached by the inscribed or circumscribed figure.
Valerio had made the gap between the method of exhaustion and the
limit idea narrower by stating directly that the approximating in-
scribed and circumscribed figures could be made to differ by less than
any magnitude; and Stevin had aided significantly in this direction
by his omission, on occasion, of one of the two approximating figures
and by his use of arithmetic sequences. It was Gregory of St. Vincent,
however, who gave perhaps the first explicit statement that an infinite
series defines in itself a magnitude which may be called the limit of
the series. "The terminus of a progression is the end of the series to
which the progression does not attain, even if continued to infinity,
but to which it can approach more closely than by any given inter-
val." 160 As an illustration, Gregory gave a line segment AK, which
was subdivided by points B,C, . . . with K as their limit and such that
the segments AB, BC, . . . were in continued proportion. Gregory,
however, lost somewhat the force of the limit concept by remarking
that "the magnitude AK is equal to the magnitude of the whole pro-
gression . . . continued to infinity; or, which is the same thing, K is
"» Wallner ("Uber die Entstehung des Grenzbegriffes," pp. 251-52) says this was the
first use of a truly infinite series: but in this he is mistaken, inasmuch as Calculator,
Oresme, and other Scholastics had employed them long before, as we have seen, in con-
nection with the latitude of forms.
m Opus geometricum, p. 55.
138 A Century of Anticipation
the terminus of the ratio AB to BC continued to infinity." 161 In other
words, AK was not defined as the sum, because K was the limit:
Gregory pointed out, rather, on the basis of geometrical intuition, that
the remark that AK is the sum, is equivalent to the statement that K
is the limit. Nevertheless, he stated more clearly than had anyone pre-
viously that an infinite series can be strictly considered as having a sum.
Gregory of St. Vincent recognized also that the paradox in the
Achilles is to be explained in terms of the limit of an infinite series.
Assuming, as had the Scholastics and Galileo, that "motion is a kind
of quantity," he asserted that the speeds of Achilles and the tortoise
must have a proportion and calculated by geometric progressions at
what point the positions of the two will coincide. 182 He failed to recog-
nize in this connection that the question of Zeno was not when or
where Achilles would overtake the tortoise, but rather how he could do it.
This appeal in the paradox to sensory experience, instead of reasoning,
was to be without doubt the chief obstacle to the development of the
calculus in terms of the limit of an infinite sequence. Although Gregory
of St. Vincent did not express himself with the rigor and clarity char-
acteristic of the nineteenth century, his work is to be kept in mind as
constituting the first attempt explicitly to formulate in a positive
sense — although still in geometrical terminology — the limit doctrine
which had been implicitly assumed by both Stevin and Valerio, as
also, probably, by Archimedes in his method of exhaustion.
Gregory of St. Vincent maintained that he had squared the circle. 153
Perhaps it is on account of this fact 154 that it has been said 166 that
Gregory received only disdain on the part of his contemporaries, his
memory being rehabilitated by Huygens and Leibniz. 156 On the other
hand, there can be no doubt that his work exerted a strong influence
on many of the mathematicians of his time. As a teacher in various
Jesuit schools, he numbered among his disciples Paul Guldin, Andreas
Tacquet, 157 Jean-Charles della Faille, 158 and others who used infinites-
» Ibid., p. 97. 152 Ibid., pp. 101-3. 1M Ibid., pp. 1099 ff.
154 Bosmans, "Gregoire de Saint- Vincent," pp. 254-55.
156 Marie, Hisloire des sciences mathematiques, III, 187.
166 Cf. Leibniz, Mathematische Schriften, V, 331-32.
157 See Kaestner, Gesckichte der Mathematik, III, 266-84, 442-49, for summaries of
Tacquet's works.
188 See Bosmans, "Le Mathematicien anversois Jean-Charles della Faille de la Com-
pagnie de Jesus."
A Century of Anticipation 139
imal considerations, particularly in the then-popular problem of de-
termining centers of gravity.
Although Guldin wrote in the old Archimedean manner, his work
has become well known for two reasons. In the first place, the so-called
Guldin theorem — that the volume of a solid of revolution is given by
the product of the rotating area and the distance through which the
center of gravity moves in one revolution 159 — has been much discussed
from the point of view of possible plagiarism from Pappus. 160 Secondly,
Guldin became, as we have seen, the chief critic of the lack of rigor in
the use of infinitesimals by Kepler and of indivisibles by Cavalieri.
From the point of view of the development of ideas, however, the work
of Tacquet was more significant than that of Guldin.
Andre (or Andreas) Tacquet resembles his contemporary, Torri-
celli, in the generality of his adoption from his predecessors of varied
infinitesimal methods. In his Cylindricorum et annularium libri IV he
gave, for example, four demonstrations of the proposition that the
volume of a sphere is equal to that of a cylindrical wedge whose base
is half a great circle of the sphere, and whose altitude is equal to the
circumference of the sphere. This theorem had been given by a number
of mathematicians since Kepler, as well as by Archimedes in the
Method, probably not then extant. Tacquet, however, after proving
the theorem in two ways by the use of inscribed and circumscribed
figures, gave two further demonstrations by indivisibles, based on the
equality of triangles and circular sections. Torricelli had himself been
satisfied with the rigor of proofs by means of indivisibles, although he
supplied alternative demonstrations for the benefit of others. Tacquet,
on the other hand, said that he did not consider that the method of
Cavalieri was to be admitted as either legitimate or geometrical. 161 He
maintained that the cylindrical wedge could not, in all strictness, be
considered as made up of triangles; nor could the sphere be regarded
168 Kepler had given a special case of this theorem in 1615, in his determination of the
volume of the tore.
wo For a bitter discussion on this point, see Smith and Miller, "Was Guldin a plagiarist?"
The editor of the recent French edition of Pappus' works, Paul Ver Eecke, exonerates
Guldin in no uncertain terms, saying that the theorem of Guldin cannot be deduced from
the form in which it is given by Pappus, and indeed that a link by inspiration between the
two is open to serious doubt. See Ver Eecke, "Le Theoreme dit de Guldin consider^ au
point de vue historique."
ltl Cylindricorum et annularium, pp. 23-24; or Opera, 3d pagination, p. 13.
140 A Century of Anticipation
as composed of circles. Neither was the wedge generated by the flux
of a triangle, nor the sphere by the motion of a circle. A geometrical
magnitude, he asserted, is made up only of homogenea, that is, parts
of like dimension — a solid of small solids, an area of small areas,
and a line of small lines — and not of heterogenea, or parts of a lower
dimension, as Cavalieri had maintained. He therefore felt that a pro-
posed magnitude was to be exhausted (a word which he undoubtedly
acquired from Gregory of St. Vincent) by inscribing homogenea within
them "as in the manner of the ancients." 162
Tacquet's criticism of the method of indivisibles and his insistence
on the use, instead, of homogenea was, of course, quite justified. Had
he tried to reconcile the two points of view by employing the word
"exhaust" in its literal sense in terms of the doctrine of limits, his
work might have served to clarify the method of indivisibles which
men continued to employ, not because they understood its significance,
but because it invariably gave them correct results. It is the more
strange that Tacquet did not do this, inasmuch as he developed the
thoughts of Gregory of St. Vincent on limits. For example, in his
Arithmeticae theoria et praxis of 1656, he explained the Achilles in
terms of geometrical progressions; 163 and in another passage he re-
marked that in a progression continued ad infinitum, in which the
terms decrease in a given proportion, the smallest term vanishes, 164
thus applying the criterion of a limit. This arithmetical work of Tac-
quet appeared almost simultaneously with the Arithmetica infinitorutn
of John Wallis, in which we shall find the notion of limit more vigor-
ously proposed. Before reviewing the ideas of Wallis, however, it will
be necessary to consider at some length the significant anticipations of
methods of the calculus made by a group of illustrious contemporary
French mathematicians — Roberval, Pascal, and Fermat — upon at
least one of whom — Pascal — Tacquet exerted an influence.
Giles Persone de Roberval held the chair of Ramus at the College
Royal, a position that depended upon supremacy in an examination
held every three years, the questions for which were propounded by
the incumbent at the time. To this fact Roberval attributed the secrecy,
with respect to his methods and results, which eventuated in his loss
m Ibid. m Arithmeticae theoria et praxis, pp. 502-3.
w "Minimus terminus evanescat." Ibid., p. 475; cf. also Bosmans, „1Andr6 Tacquet et
son traits d'arithm&ique theorique et pratique."
A Century of Anticipation 141
to other men of credit for priority. He said that he had considered at-
tentively the "divine Archimedes," and that from this study he worked
out for himself "the sublime and never to be sufficiently praised doc-
trine of the infinite." 166 Roberval seems to have worked out his method
of indivisibles between the years 1628 and 1634, 166 that is, only a few
years after Gregory of St. Vincent and Cavalieri had developed theirs
and before the work of either had been published. The almost simul-
taneous appearance of these procedures indicates how widespread was
the tendency toward infinitesimal considerations during the early
seventeenth century.
Roberval admitted no inspiration for his work other than that of
Archimedes. It is quite probable, nevertheless, that he was to some
extent influenced by Kepler, 167 and portions of his work greatly re-
semble the ideas of Stevin and Valerio; but we can point to no clear
indications of indebtedness. 168 It would be interesting to know to just
what extent the work of Stevin was known in France during the first
half of the seventeenth century. The similarity of some of the ideas
expressed by Pascal and Roberval to those found in the writings of
Stevin half a century earlier is striking; but the former scientists made
no acknowledgment of indebtedness to the engineer of Bruges, whose
works had appeared several times in Flemish, Latin, and French. 169
Roberval was without doubt familiar with the work of Cavalieri,
which he defended from carping critics; 170 but his view of indivisibles
appears to have been far less naive than that of the Italian. He said
quite clearly that in his method he did not regard a surface as really
composed of lines or a solid as made up of surfaces, but as in actuality
built up of small pieces of surfaces and solids respectively, these "infi-
nite things" being regarded "just as if they were indivisibles." 171 In his
Traiti des indivisibles, Roberval asserted that throughout the discus-
sion it was to be understood that the phrase "the infinite number of
1,6 Walker, A Study of the TraUi des Indivisibles of Roberval, pp. 15-16.
"'Ibid., pp. 142-64.
u1 Ibid., p. SI.
"■ Duhem (Les Origines de la statique, I, 290-326) asserts that Roberval knew the work
of Stevin and Valerio.
1,9 It has been suggested (see The Physical Treatises of Pascal, p. 4, n.) that Stevin
was the victim, because of his liberality of thought, of a conspiracy of silence on the part
of Catholics in the Low Countries.
lw "Divers ouvrages de M. Personier de Roberval," p. 444.
171 Walker, A Study of the TraiU des Indivisibles of Roberval, p. 16.
142 A Century of Anticipation
points" is used for the infinity of little lines which make up the whole
line, and that "the infinite number of lines" represents the infinity
of little surfaces which make up the whole surface, and so on. 172
Roberval attributed like views to Cavalieri, saying that the latter
did not really mean that a surface was made up of lines; but in this
he was, apparently, overgenerous. Influenced principally by the classic
works of Archimedes, Roberval did not recognize that in Cavalieri's
work, as in that of Galileo and Torricelli, the atomic and Scholastic
traditions had operated to modify the author's thought and result in
the method of indivisibles. Unlike Archimedes, however, Roberval
substituted the conception of infinity for the method of exhaustion,
somewhat in the manner of Gregory of St. Vincent, but without ex-
plicitly formulating the limit concept. He did, however, supply the
essential element found in our conception of the definite integral, in
that, after dividing a figure into small sections, he allowed these con-
tinually to decrease in magnitude, the work being carried out largely
arithmetically and the result being obtained by summing an infinite
series. This method, which resembles to a considerable extent the pro-
cedures of Stevin, contrasts strongly with that of the fixed indivisibles
of geometric character which is found in the work of Cavalieri.
It has been indicated that the equivalent of the theorem f%x n dx =
a H + x
had been anticipated by Cavalieri for positive integral values
n + 1
of n, and by Torricelli for rational values of n (except, of course, for
n = —1). At about the same time Roberval had arrived at this result,
perhaps on Fermat's suggestion, through investigations which bring
out nicely the somewhat different emphasis found in his work. Whereas
Cavalieri and Torricelli had proceeded on the basis of the purely geo-
metrical considerations involved in the method of exhaustion and in
the method of indivisibles, the great French mathematical triumvirate
of Roberval, Fermat, and Blaise Pascal 173 combined their interest in
the geometry of Archimedes with an enthusiasm for the theory of
numbers, and this colored their work. As a consequence, Roberval was
led to make an association between numbers and geometrical magni-
m "Divers ouvrages," pp. 249-50.
173 Their famous contemporaries and countrymen, Desargues and Descartes, in this
respect display a somewhat different spirit.
A Century of Anticipation 143
tudes, which resembles strongly that of the Pythagoreans, particularly
that of Nicomachus. A line segment, it has been remarked, Roberval
regarded as made up of an infinite number of little lines, represented
by points, which can be made to correspond to the positive integers.
If, now, we consider successively the right isosceles triangles with
sides made up of 4, 5, 6, . . . points or indivisibles respectively, the
total number of such units in the triangles will be given as follows:
The triangle of 4 is 10 = |(4) 2 + *(4)
The triangle of 5 is 15 = |(5) 2 + 1(5)
The triangle of 6 is 21 = K6) 2 + *(6)
The second term on the right in each line is half of the side, and rep-
resents the excess of the triangle over the half square. This continues
to diminish, in proportion to the first term, as the number of points and
lines is increased. Since the number of lines in a geometrical triangle
or square is infinite, the excess or half of one line, "does not enter into
consideration." 174 It is therefore clear that the triangle is half the
square. This argument is evidently roughly equivalent to that indi-
cated by the notation J^xdx = — .
Roberval continued this type of work by remarking similarly that
if the lines should follow one another in the order of the square, the
sum of all these little lines, or the points which represent them, would
be to the last, taken an equal number of times, as the pyramid is to
the prism, that is, as 1 to 3. For example, if we take a square pyramid
of dots having four on an edge, we have l 2 -+- 2 2 + 3 2 + 4 2 = 30 =
1(4)' + i(4) 2 + i(4); if there are five on an edge, we have l 2 + 2 2 +
3 a + 4 2 + 5 2 = 55 = i(5) 3 + K5) 2 + i(5); and so on. In these equa-
tions the first term on the right is one-third the cube, the second is
one-half the square, and the last is one-sixth the number of points in
the edge of the base of the pyramid. Inasmuch as the number of squares
in a geometrical cube is infinite, the last two terms are as nothing, so
»4 "Divers ouvrages," pp. 247-48.
144
A Century of Anticipation
that the sum is § the cube. 176 In the same way, the sum of the cubes is
one-fourth of the fourth power; the sum of the fourth powers is one-
fifth of the fifth power; the sum of the fifth powers is one-sixth of the
sixth power, and so on. 176 In other words, Roberval has in this manner
.» + i
indicated the equivalent of the theorem, flx n chc =
»-f 1
, for positive
integral values of n. He appears not to have given a demonstration for
other values of w, 177 as did Torricelli and Fermat.
The arguments of Roberval resemble the "demonstrations by
arithmetic" given by Stevin half a century earlier, and similar ones
of Wallis a few years after Roberval's work. 178 All of these represent
efforts to express the notion of limit, but Roberval in his work obscured
B E' F' G' H' V C
V
» ' » • P\j
i » i \ / •
x x ±--oy.---i
T — T"J^( \ — r
Lti^-<^ L ' J '
a>^^- } j J r
2"
S'
Q'
E F G H
FIGURE 17
D
the limit idea somewhat by resorting to his notion of indivisibles.
Instead of drawing his conclusion from the limits of the arithmetic
sequences involved, he had recourse, as did most of his contempor-
aries, to geometrical intuition. Making use of the Pythagorean and
Nicomachean association of numbers with geometrical points, he re-
marked that since "the side has no ratio to the cube, . . . adding or
subtracting a single square has no effect." 179 Intuition of this type led,
175 Ibid., p. 248. "« Ibid., pp. 248-49. m Zeuthen, "Notes," 1895, p. 43.
178 Walker (A Study of the Traitt des Indivisibles of Roberval, p. 165) makes no mention
of Stevin in this respect, incorrectly representing Roberval's work as the first of its type.
The statement that the idea of an arithmetic limit appeared for the first time in the
seventeenth century (ibid., p. 35) is, of course, not accurate, inasmuch as Stevin's arith-
metical work in limits appeared in 1586.
179 "Divers ouvrages," p. 249.
A Century of Anticipation 145
through the neglect of infinitesimals of higher order, to the differential
calculus of Leibniz, rather than to the method of limits which Newton
suggested and which triumphed in the end.
Roberval successfully applied this quasi-arithmetical method of
indivisibles to varied problems in quadratures. Typical of these is his
quadrature of the parabola. 180 The procedure which he employed is
quite different from any given by Torricelli in his twenty-one quad-
ratures of the parabola. Roberval's method resembles, rather, Stevin's
demonstration by numbers, reinforced by intuition of indivisibles. Let
AE = 1, AF = 2, AG = 3, . . . (fig. 17). Then from the definition of
EL AE?
the parabola we know that — = , and similarly for the other
v FM AF 2
points of division. Hence
area ADC _ AE(EL + FM + GN + . . .)
area ABCD ~ AD • DC
_ A E(AE? + AF* + AG 2 + . . ■)
AD AD*
- I 2 + 2 2 + 3 2 + . . . + AD 2 _ x
AD- AD* ~ 3
Roberval by similar methods found the areas under other curves,
such as parabolas of higher degrees, the hyperbola, the cycloid, and
the sine curve, as well as various volumes and centers of gravity as-
sociated with these. He may in this connection have anticipated Torri-
celli in finding the volumes of infinitely long solids. He also used an in-
genious transformation of one figure into another which came to be
spoken of as the method of Robervallian lines and which resembled
the ductus plant in planum of Gregory of St. Vincent. Such transforma-
tions played a large part in the geometry of the seventeenth century
because of the lack of a facile method for handling curves whose equa-
tions involved radicals, but after the development of the calculus these
lost their popularity, as well as their significance.
Roberval in his work showed a remarkable flexibility, using divers
infinitesimal elements, such as triangles, parallelograms, parallele-
pipeds, cylinders, and concentric cylindrical shells. Throughout it all,
*>Ibid., pp. 256-59.
146 A Century of Anticipation
the idea of limits is implied, but is concealed under the terminology
of Roberval's method of indivisibles. An adumbration of the method
of limits is indicated also by the manner in which Roberval reconciled
the demonstrations by indivisibles with those of the ancient geometers.
First Roberval showed that the unknown quantity lies between in-
scribed and circumscribed figures which differ "by less than every
known quantity proposed." Then he showed that the quantity in
question bore to the circumscribed figure a ratio less, and to the in-
scribed figure a ratio more, than the proposed ratio. Finally Roberval
proved the proposition by the application of a general lemma: "If
there is a true ratio R : S and two quantities A and B such that for a
small (quantity) added to A, then this sum has to B a greater ratio
then R: S and for a small (quantity) subtracted from A, the remainder
has to B a ratio less than R: S; then I say that A : B as R: S." 181 This
form of argument, resembling strikingly the corresponding propositions
of Valerio (with whose work he may have been familiar), is equivalent
to a statement that the limit of a quotient of variables is equal to the
quotient of their respective limits.
In Roberval's propositions on indivisibles one recognizes numerous
anticipations of the integral calculus, some of which are equivalent to
the determination of definite integrals of algebraic and trignometric
functions. Roberval was concerned, as well, with problems of the differ-
ential calculus — for he developed a method of tangents so much like
that of Torricelli that charges of plagiarism arose. 182 He regarded every
curve as the path of a moving point, and accepted as an axiom that
the direction of motion is also that of the tangent. 183 By looking upon
the motion of the point as made up of two component movements,
he found the tangent by determining the resultant of these. Thus he
found the tangent to the parabola by making use of the fact that,
since this curve is the locus of points equidistant from the focus and
the directrix, it may be regarded as generated by a point moving with
a compound motion made up of a uniform motion of translation away
from the directrix and an equal uniform radial motion away from the
focus. By the parallelogram (in this case a rhombus) of velocities it is
181 Walker, A Study of the Traiti des Indivisibles of Roberval, pp. 38-39.
m "Divers ouvrages," pp. 436-78; Walker, op. cit., pp. 142-64. Moritz Cantor (Vorle-
sungen, II, 808-14) concludes that the charges are not substantiated.
ms "Divers ouvrages," p. 24.
A Century of Anticipation 147
therefore determined that the resultant velocity — and consequently
the tangent to the parabola at any point— will be in the direction of
the bisector of the angle between the focal radius at the point and the
perpendicular from the point to the directrix. This direction being
known, the tangent can be drawn. 184 The motions involved here are
different from those used for the same curve by Torricelli, but the
underlying idea of the composition of movements is essentially the
same. The method is, of course, subject to the difficulty that one must
in some way discover the laws of motion before one can determine the
tangent. Roberval appears, from his correspondence with Fermat in
1636, to have had another method of tangents which proceeded ana-
lytically and which he says was connected with the problem of quad-
ratures. 186 This might have been significant in the history of the
calculus, but it apparently was lost.
It is difficult to determine the extent of Roberval!s influence on con-
temporary mathematicians inasmuch as his Traite des indivisibles was
not published until 1693 — that is, only after the calculus itself had
been made known. It is very probable, however, that he had a strong
influence upon Pascal the younger, whose father, Etienne Pascal, was
a close friend of Roberval.
Blaise Pascal in a sense represents the highest development of the
method of infinitesimals carried out under the traditions of classical
geometry. He was not so much a creative genius as a mathematician,
scientist, and philosopher, with a remarkable flair for clarifying ideas
which had been somewhat vaguely set forth by others, and for sup-
plying these with a reasonable basis. 186 This penchant of Pascal's is
well illustrated in science by his lucid organization of the principles of
hydrostatics; in mathematics one sees it in his exposition of the nature
of infinitesimals, in which, to be sure, one perceives also a touch of his
characteristic mystical turn.
Pascal was not a professional geometer, and as a result his geo-
metrical work was accomplished in two periods which were separated
by an interval of mathematical inactivity (from 1654 to 1658) during
which he devoted his interests to theology. These two periods, more-
over, are characterized by somewhat different views as to the nature
184 Ibid., pp. 24r-26. » See Moritz Cantor, Vorlesungen, II, 812.
m Bosnians, "Sur l'oeuvre mathematique de Blaise Pascal.' 5
148 A Century of Anticipation
of infinitesimals. Pascal had two predominating interests in mathe-
matics: geometry and the theory of numbers. Toward the end of the
first of his two periods of,mathematical work the latter was dominant,
and at this time he applied the theory of infinitesimals to his work on
the arithmetic triangle. Although this is usually called Pascal's tri-
angle, the ordering of binomial coefficients had been known to Stifel
long before. 187
In this connection he enunciated, in the Potestatum numericarum
summa of 1654, the theorem on the integral of x n , which we have met
with in the work of Cavalieri, Torricelli, and Roberval. Pascal's dem-
onstration of this 188 is derived not from classic geometrical propositions
alone, but from an examination of the figurate numbers represented in
the arithmetic triangle — a form of proof suggestive of that of Roberval,
which appears not to have been generally known at Paris at the time. 189
In the arithmetic triangle the numbers in the first row (or column) may
be considered units or points making up a line. Those in the second
row represent the sums of the numbers in the first row and may be
11111..
12 3 4..
13 6..
14..
1 . .
considered therefore as the sums of points or units — that is, as lines.
The numbers in the third row, which are in turn the sums of those in
the second row, may therefore be considered as the sums of lines, that
is, as triangles. The numbers in the fourth row similarly represent
pyramids. Geometrical intuition now fails, but one can continue by
analogy. 190
From such geometrical considerations and from the numerical re-
lationships within the triangle, Pascal was led immediately to con-
sider the sums of powers of the positive integers. Recalling the results
by the geometrical procedures of the ancients for the sums of the
187 See Bosmans, "Note historique sur le triangle arithme'tique de Pascal."
m (Euvres, III, 346-67, 433-593. "» Zeuthen, "Notes," 1895, p. 43.
w Cf. Bosmans, "Sur l'interpr6tation geome'trique, donnee par Pascal a l'espace a
quatre dimensions."
A Century of Anticipation 149
squares and of the cubes, he recognized that these were not imme-
diately applicable to powers of higher degree. Pascal, however, de-
veloped a general arithmetical method for determining the sum, not
only in the case of terms which are integral powers (of the same de-
gree) of the first N natural numbers, but also for powers (of the same
degree) of any integers in arithmetic progression. Pascal expressed his
result rhetorically, but this may be given in symbolic form by the
equation:
■ + , Ci<te w + " + 1 C«#Z ( — I) + . . . + n + 1 C„J w S (1) = (a + Nd) n + l
-a" + l - NdT + l ,
where a is the first term of the progression, d the common difference,
N the number of terms, n the degree of the powers in question, " + ! Q
the number in the (i + l)st column and the (n — i + 2)nd row in
Pascal's triangle, and S 0) the sum of the jth powers of the terms of the
progression.
As Pascal remarked, it will be obvious to any one who is at all fa-
miliar with the doctrine of indivisibles that this result can be applied
to the determination of curvilinear areas. In order to find the area
under the curve y = x n , for example, the surface in question is to be
regarded as the sum of ordinates which are the nth powers of abscissae
chosen in arithmetic progression (with first term zero and with common
difference equal to unity), of which in this case there will be an infinite
number. Moreover, a single point adds nothing to the length of a line:
nor does the addition of a line to a surface cause any difference in area,
for the former is an indivisible with respect to the latter. Or, speaking
arithmetically, roots do not figure in a ratio of squares, nor squares in
a ratio of cubes, and so on. 191 The rule given above consequently be-
comes, on calling the greatest abscissa b and on neglecting as zero the
terms of lower order, {n + l)S (w) = b n + \ In a general sense this is,
b n + l
of course, the equivalent of the expression $&*dx = .
n-f 1
Or, translating this into the terminology often used at the time, the sum
of lines in a triangle is half the square of the longest; the sum of the
squares of the lines is one- third the cube; the sum of the cubes is one-
fourth the "square-square," and so on.
The essential point in Pascal's demonstration is the omission of
*» (Euvres, HI, 366-67.
150 A Century of Anticipation
terms of lower dimension. This type of argument has frequently been
attributed to Cavalieri, 192 but there appears to be no basis for such a
view. Cavalieri 's method was based upon a strict correspondence of the
indivisibles in two figures, and there were no unpaired or omitted ele-
ments. The method of dropping terms seems to have entered in the
work of Roberval and Pascal through the association of the indivisible
of geometry with arithmetic and the theory of numbers. The geo-
metrical intuition of indivisibles of lower dimension was, in their work,
carried over into arithmetic to justify the neglect of certain terms of
lower degree. Pascal went so far as to compare the indivisible of ge-
ometry with the zero of arithmetic, much as Euler later regarded the
differentials of the calculus as nothing but zeros.
This neglect of quantities, as found in Pascal, has been character-
ized 193 as the basic principle of the differential calculus. Such a desig-
nation is indeed misleading, for the subject is no longer explained in
terms of the omission of fixed infinitesimals. Nevertheless, the work of
Pascal exerted perhaps the strongest influence in shaping the views
of Leibniz, who adopted into his calculus as fundamental the doctrine
that "differences" of higher order could be neglected. Newton also oc-
casionally lapsed into this type of argument in dropping out of the cal-
culation "moments" which did not add significantly to the result. For
almost two centuries mathematicians tried to justify such procedures,
but in the end the basis of analysis was found, not in these, but rather
in the method of limits toward which the geometrical method of ex-
haustion and arithmetical modifications of this by Stevin, Tacquet,
Roberval, and others had pointed.
In answering the objections of those of his contemporaries who held
that the omission of infinitely small quantities constituted a violation
of common sense, Pascal had recourse to a favorite theme — that the
heart intervenes to make this work clear. In this case what is necessary
is the "esprit de finesse," or intuition, rather than the "esprit de geo-
metric," or logical thought, much as the action of grace, as well as
physical experience, is above reason. In this respect the paradoxes of
geometry are to be compared to the apparent absurdities of Christi-
m Ball, History of Mathematics, p. 249; Cajori, History of Mathematics, p. 161; Marie,
Histoire des sciences mathematiques, IV, 72; Milhaud, "Note sur les origines du calcul
infinitesimal," p. 35.
"• Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 120.
A Century of Anticipation 151
anity, the indivisible being to geometrical configurations as our justice
to God's. 194
The mysticism which Pascal often displayed in his attitude toward
the infinitesimal does not appear in all of his work. Particularly in the
later period of his mathematical activity — in which his interest cen-
tered about the cycloid, the curve which Montucla called "la pomme
de discorde" 195 because it engendered so many quarrels with respect to
priority — his view appears to be modified. In connection with problems
such as those in his Traite des sinus du quart de cercle of 1659 in which
he balanced elements as Archimedes had done in his mechanical
method, he used the language of infinitesimals in speaking of the sum
of all the ordinates; but he added that one need not fear to do this,
inasmuch as what is really meant is the sum of arbitrarily small rec-
tangles. 196 In his later numerical demonstrations also Pascal sought to
avoid arguments based upon the neglect of infinitely small quantities.
The Aristotelian view had denied the existence in the realm of number
of the infinitely small, while admitting, as a potentiality, the infinitely
great. Pascal, on the contrary, maintained in De Vesprit geometrique
that in the sphere of number the infinitely great and small are com-
plementary. Corresponding to every large number, such as 100,000,
there existed a small one, the reciprocal i o*ooo so that the existence
of the indefinitely large implied that of the indefinitely small. Number,
he held, was as much subject to the two infinities — in greatness and in
smallness — as were such other undefined primitive terms in geometry
as time, motion, and space. 197 The contrast between discrete and con-
tinuous magnitudes was not so great as Aristole had felt, and was, in
fact, vanishing with the spread of analytic methods in geometry.
The change in Pascal to a clear point of view with respect to infini-
tesimals may have been the result of friendship with Roberval, who had
said that Cavalieri did not really think of indivisibles as lines. It may
equally well have come from Pascal's reading of Tacquet's Cylindri-
corum et annularium, 19 * in which the author denied the validity of con-
cluding anything about the ratio of surfaces from the ratio of their
indivisibles, or lines. Tacquet had been particularly emphatic in deny-
m See (Euvres, XII, 9, XIII, 141-55.
m Montucla, Histoire des mathematiques, II, 52.
"• (Euvres, IX, 60-76. IW (Euvres, IX, 247, 253, 256, 268.
i» Bosnians, "La Notion des indivisibles chez Blaise Pascal."
152 A Century of Anticipation
ing that a configuration could be thought of as composed of heterogenea,
or elements of a lower dimensionality. Pascal was in general agreement
with him on the question of homogeneity, but his view of the transition
from the finite to the infinite was different. Tacquet was inclined to-
ward the limit idea of Gregory of St. Vincent, although he preferred
to avoid the difficulty by returning to the clarity afforded by the
method of exhaustion.
Pascal, on the other hand, looked upon the infinitely large and the
infinitely small as mysteries — something which nature has proposed
to man, not to understand, but to admire. 199 Furthermore, Tacquet
had made use of the limits of infinite series, as had also Stevin and
Roberval. The work of Pascal, however, was carried out in connection
with the older theory of numbers and classical geometry of which he
considered his method an elaboration. The newer analytic procedures
of Fermat and Descartes did not appeal to him, and for them he sub-
stituted a remarkable facility in the manipulation of geometric trans-
formations, similar to those of Gregory of St. Vincent and Roberval.
Through these he related the figurate sums of his number theory to
problems in the synthetic geometry of continuous magnitudes and an-
ticipated numerous results of the integral calculus, including the
equivalent of integration by parts. His underestimation of the value
of the algebraic and the analytic viewpoints may have been responsible
not only for his inability to define the central and unifying concept of
the integral calculus — that of a limit of a sum — but also for his failure
to recognize the inverse nature of the problems of quadratures and
tangents.
The idea and figure of what is now called the differential triangle had
appeared on several occasions before the time of Pascal, and even as
early as 1624. Snell, in his Tiphys Batavus, had thought of a small
spherical surface bounded by a loxodrome, a circle of latitude, and a
meridian of longitude as equivalent to a plane right triangle. 200 Numer-
ous diagrams somewhat resembling the differential triangle are to be
found in the geometrical works of the middle seventeenth century,
such as those in the De infinitis hyperbolis of Torricelli and the Traite
des indivisibles of Roberval, with which Pascal may have been familiar.
"» (Euvres, DC, 268.
100 Aubry, "Sur Phistoire du calcul infinit6simal entre les ann6es 1620 et 1660," p. 84.
A Century of Anticipation
153
In all of these, however, the significance of the quotient of two sides of
the triangle for the determination of tangents appears to have escaped
emphasis. It was much the same with Pascal. In connection with a
diagram (see fig. 18) from his Traite des sinus du quart de cercle of 1659,
he remarked that AD is to DI as EE is to RR or EK, and that for
small intervals the arc may be substituted for the tangent. Pascal made
use of these lemmas to determine the sum of the sines (ordinates) of a
portion of the curve, that is, the area under this portion. If Pascal had
at this point only been more interested in arithmetic considerations
and in the problem of tangents, he might have anticipated the im-
portant concept of the limit of a quotient and have discovered the
R I R
FIGURE 18
significance of this for the determination of both tangents and quad-
ratures. Had he done this, he would have hit upon the crucial point in
the calculus some seven years before Newton and about fourteen years
before Leibniz. The latter, who later, as we shall see, made use of this
very diagram to establish his infinitesimal calculus, said, in 1703, in a
letter to James Bernoulli, that sometimes Pascal seemed to have had
a bandage over his eyes. 201 This apparent lack of imagination was very
likely the result of a predilection for the classical, such as later re-
strained the scientist Huygens also from making full use of the new
procedures.
m See Leibniz, The Early Mathematical Manuscripts, pp. 15-16, and Mathematische
Schrijtm, HI, 72-73, n.
154 A Century of Anticipation
Pierre de Fermat, the friend of Pascal and perhaps the greatest
French mathematician of the century, possessed a singular erudition
and displayed an enthusiastic interest in Greek and Latin philology.
This led him to study carefully such classic mathematical works as
those of Archimedes, Apollonius, and Diophantus. The influence of
the first of these three had been very strong for almost two centuries,
but in the work of Viete and of Fermat we have evidence of the im-
pression made by the other two also, as well as by the Arabic and
Italian development of algebra. Viete realized the facility to be gained
in the handling of geometric problems by their reduction to the solu-
tion of algebraic equations, a procedure which he therefore followed
whenever possible. 202 Viete's equations betrayed their origin in ge-
ometry, in that he was always careful to have them all homogeneous;
but his work was nevertheless, in a sense, an inversion of the Greek
view, in accordance with which algebraic equations were reduced to
geometric constructions for purposes of solution.
Fermat was familiar with the methods of Viete and developed them
into an analytic geometry at about the time that Descartes was pre-
paring his famous Geometrie of 1637. The work of Fermat and Des-
cartes went much further than did either the algebraic solution of
geometric problems by Viete or the graphical representation of vari-
ables by Oresme, for it associated with each curve an equation in
which are implied all of the properties of the curve. This recognition,
which Fermat expressed in calling the equation the "specific property"
of the curve, constitutes the basic discovery of analytic geometry.
Although in publication Fermat was anticipated by Descartes, he far
outdid his rival in the application of the new point of view to the
problems of infinitesimal analysis, which the books of Kepler and
Cavalieri had popularized.
All of the anticipations of methods of the calculus which we have so
far considered were related to geometry. Infinite series had sometimes
been employed, but they were derived from the geometrical represen-
tation of the problem. Infinitesimal lines, surfaces, and solids had been
used, but not infinitesimal numbers. Aristotle had denied the infinitely
small in arithmetic, for the obvious reason that since number was a
202 For an unusually extensive account of this work, see Marie, Histoire des sciences
matktmotiques, III, 27-65.
A Century of Anticipation 155
collection of unities, no number could be smaller than one. As a result
of the algebra and the analytic geometry of the sixteenth and seven-
teenth centuries, this attitude had been modified, as has already been
seen in the case of Pascal. The present-day view of the symbols enter-
ing into an equation is that they represent, in general, continuous
variables; for Fermat and Descartes, however, they represented in-
determinate constants 203 to which line segments could be associated,
the tacit assumption being made that to every segment there corre-
sponded some number. To such a view there was nothing incongruous
in the idea of infinitesimal constants or numbers, since they would
correspond to the geometrical infinitesimals which were being used so
successfully. These numerical infinitesimals arose first through some
interesting problems considered by Fermat.
Pappus had spoken of a "minima et singularis proportio," and this
led Fermat to dwell on the fact, as he explained in a letter of 1643,
that in a problem which in general has two solutions, the maximum or
minimum value gives but a single solution. 204 Thus if a line of length
a is divided by a point P into two parts, x and a — x, there are in
general two positions of P which will make the area of the rectangle on
x and a — x a given quantity, A. For the maximum area, however,
there is only one position, the mid-point.
From this fact Fermat was led to formulate his remarkably ingenious
and fruitful method for determining maximum and minimum values.
His method first appeared in an article in 1638; but Fermat said that
the discovery went back some eight or ten years previous. 205 The argu-
ment in this is as follows: Given a line segment of length a, mark off
from one end a distance x. The area on the segments x and a — x will
then be A = x(a — x). If instead of the distance x one were to mark
off the distance x + E, the area would be A = (x + E) (a — x — E).
For the maximum area the two values will be the same, from Pappus'
observation, and the points x and x -f- E will coincide. Consequently,
203 Wallner, "Entwickelungsgeschichtliche Momente bei Entstehung der Infmitesimal-
rechnung," p. 119.
204 See Giovannozzi, "Pierre Fermat. Una lettera inedita."
205 See Wieleitner, "Bermerkungen zu Fermats Methode der Aufsuchung von Extremen-
werten." In a letter to Roberval, written in 1636, Fermat said that in 1629 he was in pos-
session of his method of maxima and minima. See Paul Tannery, "Sur la date des princi-
pales d6couvertes de Fermat"; cf. also Henry, "Recherches sur les manuscrits de Pierre
de Fermat."
156 A Century of Anticipation
setting the two values of A equal to each other and letting E vanish,
. . . a
the result is x = -. z06
2
The procedure which Fermat here employed is almost precisely that
now given in the differential calculus, except that the symbol Ax (or
occasionally h) is substituted for E. In his work there appeared, for
perhaps the first time, the idea which has become basic in such prob-
lems — that of changing the variable slightly and then letting this
change vanish. However, the reasoning by which Fermat supported
his method is far less clear than that given at the present time. Modern
analysis makes use of the concept of the limit, as the change Ax ap-
proaches zero. Fermat, however, seems to have interpreted the opera-
tion as one in which E vanishes in the sense of actually being zero. For
this reason, as Berkeley remarked in the following century, it is diffi-
cult to see by what right he took the positions x and x + E to be dif-
ferent and yet in the end said that they coincide. Fermat's argument
has frequently been interpreted in terms of the limit concept, 207 in
which E is to be regarded as a variable quantity approaching zero.
Fermat, however, does not appear to have thought of it in this way. 208
In fact the function concept and the idea of symbols as representing
variables does not seem to enter into the work of any mathematician
of the time.
In answering criticisms of his method, Fermat presented a statement
of his reasoning which appears to link it with the remarks of Oresme
and Kepler on the change at a maximum point. 209 He justified the
equating of the two values of A by remarking that at a maximum
point they are not really equal but they should be equal. He therefore
formed the pseudo-equality 210 which became equality on letting E be
zero. From this it is clear that he was thinking in terms of equations
and the infinitely small, rather than of functions and the limit concept.
"• Fermat, (Euvres, I, 133-34, 147-51; III, 121-22; Supplement, pp. 120-25. Cf. also
Voss, "Calcul differentiel," p. 246.
m Duhamel, "Memoire sur la m^thode des maxima et minima de Fermat, et sur les
methodes des tangentes de Fermat et Descartes"; cf. also Mansion, "M6thode, dite de
Fermat, pour la recherche des maxima et minima.' '
108 See Wallner, "Entwickelungsgeschichtliche Momente," pp. 122-23; cf. also Paul
Tannery, Notions kistoriques, p. 344.
** Paul Tannery, however, thinks Fermat borrowed nothing from Kepler, whose works
he probably had not read. See his Review of Vivanti, II concetto d'infinitesimo, p. 232.
210 "Adaequalitas." See (Euvres, I, 133-79, for his justification.
A Century of Anticipation
157
None the less, the method worked so beautifully that it found a ready
acceptance among mathematicians. As a result, infinitesimals were un-
critically introduced into analysis, to become firmly intrenched as the
basis of the subject for about two centuries before giving way, as the
fundamental concept of the calculus, to the rigorously defined notion
of the derivative. Even now the subject is generally known as the
"infinitesimal calculus," in spite of the fact that the infinitesimal, while
of great pragmatic value in adding to the facility of manipulation of
the subject in exercises, is logically secondary and even unnecessary.
Fermat was led by the success of his method to apply it, about 1636,
to the determination of tangents to curves. This he did as follows: Let
the curve be a parabola (fig. 19). Then from the "specific property" of
FIGURE 19
the curve it is clear that if we set OQ = a, VQ = b, and QQ' = E, we
shall have
>
Torricelli, in his work on the tangents to
b + E (a + E)*
parabolas, had frequently set down such inequalities. 211 However,
whereas Torricelli had made use of arguments by a reductio ad ab-
surdum, Fermat's characteristic procedure resembles more closely the
method of limiting values. Inasmuch as for small values of E the point
P' may be regarded as practically on the curve as well as on the tan-
gent line, the inequality becomes, as in the method for maximum
values, a pseudo-equality. By allowing E to vanish, this pseudo-
equality becomes a true equality and gives the desired result, a =
2b. m
** See Opere, I (Part 2), 304 ff., 315 ff. m (Euvres, I, 134-36; III, 122-23.
158 A Century of Anticipation
The method of tangents Fermat believed to be an application of his
method for maxima, but he was unable to explain what quantity he
was maximizing. Descartes naturally supposed that it was the length
of the line from the curve to a fixed point O on the axis of the parabola.
However, on carrying out Fermat's method on this assumption, the
result he obtained was, of course, different from that of Fermat. What
Descartes had really found was the normal to the curve — that is, the
minimum distance from a point on the axis to the curve. This would
have furnished an excellent illustration of the method for determining
a maximum or a minimum value, but inasmuch as Fermat had not
given a rule for distinguishing maxima from minima, neither he nor
Descartes recognized it as such. Descartes simply concluded that al-
though the result Fermat had obtained was correct, the method was
not generally applicable.
Perhaps as the result of the unnecessarily bitter criticism and super-
cilious attitude of Descartes, Fermat later modified his explanation for
determining tangents. 213 Instead of interpreting the method in terms
of maxima, he said that the point P' was indifferently taken as on the
curve or on the tangent line. Then after forming the pseudo-equality,
the quantity E was to vanish, to give the desired result. This procedure
is strictly comparable to that now employed in the calculus, the theo-
retical justification of which is given in terms of limits; but the ex-
planation of Fermat resembles rather the neglect of infinitesimals to
be found in the work of Leibniz. It is also strikingly suggestive of the
doctrine of perfect and imperfect equations, presented almost two
hundred years later by Carnot in his attempted concordance of the
conflicting views of the calculus then prevailing.
Fermat applied analogous considerations to the problem of deter-
mining the center of gravity of a segment of a paraboloid, again under
the misapprehension that he was employing the method of maxima
and minima. In this he let the center of gravity, O, of the segment be
a units from the vertex. On decreasing the altitude, h, of the segment
by E, the center of gravity is changed. Fermat, however, knew from
a number of lemmas that the distances of the centers of gravity of the
two segments are proportional to the altitudes and that the volumes
of the segments are to each other as the squares of the altitudes. By
* u Duhamel, "M6moire sur la m6thode des maxima et minima de Fermat/ 9 pp. 310-16.
A Century of Anticipation 159
taking moments about O, he was able to make use of these facts to set
up a pseudo-equality involving a, h, and E. In accordance with his
general principle, he allowed E to vanish and obtained the result a —
p. 214
The determination of the center of gravity of the paraboloidal seg-
ment did not constitute a new result. Archimedes had calculated this
some nineteen hundred years earlier in the Method, and Commandino
and Maurolycus had rediscovered it only a century before. Neverthe-
less, this exercise of Fermat is significant in the history of the calculus
as the first determination of the center of gravity by means of methods
equivalent to those of the differential calculus, instead of by means of
a summation resembling those of the integral calculus. Fermat's friend
Roberval was astonished that one should be able to obtain by means of
this maximum and minimum method a result which had generally been
derived from summation considerations. The integral calculus is, of
course, implied in the lemmas which Fermat employed in this con-
nection, and the method of maxima served somewhat indirectly to
determine simply the value of the constant of proportionality entering
into these. Nevertheless, this theorem might have led Fermat to a
recognition of the significance of the inverse nature of summation and
tangent problems. That this escaped him is the more strange, in that
he developed remarkable procedures for the determination of quad-
ratures as well as for tangents.
a n + i
The equivalents of what we express as fffidx = had appeared
n + 1
in various forms in the work of Cavalieri, Torricelli, Roberval, and
Pascal. Fermat also gave demonstrations of this rule — in fact, he may
well have anticipated all the others in this respect — one of which is
strikingly different from those given earlier. 216 In his earlier investiga-
tions, about 1636, he appears to have made use of the inequalities
HI J. i
1"» + 2 m + 3 m + ...+«"* > > r + 2 m + 3 M + ... + (»- l) w
m + 1
to establish the result for positive integral values of n. This constituted
a generalization of the inequalities of Archimedes which was known also
» (Euvres, I, 136-39; III, 124-26.
** For an excellent exposition of this method of quadratures, see Zeuthen, "Notes,"
1895, pp. 37-80.
160 A Century of Anticipation
to Roberval. Fermat may also have given a proof, based upon the for-
mation of figurate numbers, similar to that of Pascal. 216 However, be-
fore 1644 he had found the quadratures, cubatures, and centers of
gravity of the "parabolas" of fractional degree a m y n = b n x m , n7 curves
which he seems to have been the first to propose, but which were in-
vestigated also by Cavalieri, Torricelli, Roberval, and Pascal. Fermat,
therefore, probably possessed as early as that date a general proof of
the theorem for rational fractional values as well, although the defin-
itive redaction of this was not made until 1657. 218
In this connection, Fermat's procedure constitutes a generalization
of one found in the Opus geometricum of Gregory of St. Vincent, al-
though Fermat may not have known of this work, for he here mentions
only Archimedes. Gregory had shown that if along the horizontal
asymptote of a rectangular hyperbola points are marked off whose
distances from the center are in continued proportion, and if at these
points ordinates are erected to the curve, then the areas intercepted
between these are equal. 219 Fermat modified this process in such a way
that it could be applied to both the general fractional hyperbolas and
p
parabolas. In finding the area under y = x 9 , 220 for example, from to
x, he would take points on the axis with abscissas x, ex, e 2 x, . . . where
e < 1 (fig. 20). Then erecting the ordinates at these points, the areas of
the rectangles constructed upon successive ordinates will form an in-
finite geometric progression. As Gregory of St. Vincent and Tacquet
had earlier found the sums of such progressions, so here Fermat de-
t±il \ - e \
termined the sum of the rectangles as x 9 j p + q J. In order
v - « ' I
to find the area under the curve, however, one must have not only an
infinite number of such rectangles, but the area of each must be in-
finitely small. This can be brought about by setting e = 1. Before
218 Ibid., pp. 42-43.
217 Mersenne, Cogitata physico-mathematica; see preface to Tractatus meckanicus. See also
(Euvres de Fermat, I, 195-98.
a8 Zeuthen, "Notes," 1895, pp. 44 ff.
219 Opus geometricum, Proposition CIX, p. 586.
220 The notation of Fermat has been here slightly modified to make the meaning more
clear. The equations of Fermat retained the homogeneous character found in Viete.
A Century of Anticipation
161
doing this, however, Fermat made the substitution e = E? in order to
evaluate the indeterminate form. The sum then becomes
t±s(\ _ &
1 - E p
+ 9
P+9
= X
(1 -£)(! +£ + £* + ... +£ g " 1 )
(1 - £)(1 + £ + & + • • • + E p+q ~ l )
p_±j
gx 9
When e approaches 1, E does likewise, and the sum is then ,
P +9
which is the area under the curve. By taking e > 1, Fermat applied
the same method to the fractional hyperbolas also, finding the area
under these from any abscissa to infinity. 221
FIGURE 20
In these quadratures we see most of the essential aspects of the defi-
nite integral — the division of the area under the curve into small ele-
ments of area, the approximate numerical determination of the sum of
these by means of rectangles and the analytic equation of the curve,
and finally an attempt by Fermat to express the equivalent of what
we would call the limit of this sum, as the number of elements is in-
definitely increased and as the area of each becomes indefinitely small.
One is almost tempted to say that Fermat recognized all the aspects
except that of the integral itself; that is, he did not recognize the op-
eration involved as significant in itself. The procedure was for him, as
» (Euvres, I, 255-88; III, 216-40.
162 A Century of Anticipation
it had been for all of his predecessors, simply that of finding a quad-
rature — of answering a specific geometrical question. Only with
Newton and Leibniz were the processes involved in infinitesimal con-
siderations recognized as constituting operations, independent of any
geometrical or physical considerations, to which characteristic names
were applied. 222
That a curvilinear area, such as those under Fermat's parabolas and
hyperbolas, could be equal to one bounded only by straight lines had
been known in antiquity. It had long been held impossible, however,
that a curved line could be exactly equal in length to a straight line, 223
and Fermat shared this view with a number of his contemporaries.
Sluse and Pascal in this connection expressed admiration for the order
of nature, which refused to allow a curve to equal a line. 224 Gregory of
St. Vincent, Torricelli, Roberval, and Pascal, nevertheless, had by
infinitesimal and kinematic means compared the arcs of spirals with
those of parabolas. Then, shortly before 1660, there suddenly appeared
a number of rectifications of curved lines by William Neil, Christopher
Wren, Heinrich van Heuraet, John Wallis, and others. 226
These were, in general, based upon approximations to the curve by
means of polygons, followed by applications of infinitesimal or limit
considerations. Upon hearing of them, Fermat himself carried out a
rectification of the semicubical parabola. His procedure in this con-
nection is typical of his general approach and indicates well the inter-
relation of the various aspects of his work. For any point P on the
curve with abscissa OQ — a and ordinate PQ = b, the subtangent,
TQ = c, is known by his tangent method to be c = \a (fig. 21). If
then an ordinate P'Q' to the tangent line is erected at a distance E
from the ordinate PQ, the segment PP' is known in terms of a and E.
[9a
For the curve ky 2 = x?, this is PP' = EiJ — + 1. But the point
m Simon appears to be overenthusiastic in saying of this work ("Geschichte und Phil-
osophic der Differentialrechnung," p. 119): "F. hatte auch bereits in ahnlicher Weise wie
spater Riemann den Begriff des bestimmten Integrals erfasst bei der Berechnung von
*
fx?dx. Hier ist Grenziibergang, hier ist Bestimmung des Werthes -, bier is volliges Be-
wusstsein des continuitats gesetzes."
223 Kaestner, Gesckichte der Mathematik, I, 498; III, 283.
224 See Zeuthen, "Notes," 1895, pp. 73-76; and Pascal, (Euvres, VIII, 145, IX, 201.
m Moritz Cantor, Vorlesungen, II, 827 ff .
A Century of Anticipation
163
P', for small values of E, may be regarded as on the curve as well as
on the tangent line, so that the length of the curve can be thought of
as the sum of segments such as PP' . The sum of these segments, in
9x
turn, can be taken as the area under the parabola y 2 = -— + 1. Inas-
4k
much as the quadrature of this is known, the length of the curve is
determined. 226
It is surprising that Fermat, who used his method of maxima and
minima for finding centers of gravity, who reduced a problem of recti-
fication which involved tangents to a question of quadratures, who
O T Q
FIGURE 21
used infinitesimals geometrically and analytically in such a wide
variety of problems, should have missed seeing, as Pascal had also,
the fundamental connection between the two types of questions. Be-
cause these two men did not see this, in problems in which integrations
by parts would now be employed they had recourse to clever geo-
metrical transformations. Fermat, in his problems, made use of dia-
grams which are much the same as the one of Pascal's which Leibniz
later found so suggestive of his differential triangle, and yet he did not
perceive their deeper significance. Had Fermat only observed more
"• For Fermat's rectification, see (Euvres, I, 211-54; m, 181-215.
164 A Century of Anticipation
closely the results for the tangents and the quadratures of his parab-
olas and hyperbolas, he might have discovered the fundamental
theorem of the calculus and have become, what he has sometimes been
unwarrantedly called, the "true inventor of the calculus." 227
Fermat, of course, realized in a sense that the two types of problems
had an inverse relationship. That he did not pursue this thought
further may well have been due to the fact that he thought of his
work simply as the solution of geometrical problems and not as rep-
resenting a type of argument significant in itself. His methods of
maxima and minima, of tangents, and of quadratures he regarded as
constituting characteristic approaches to these questions, rather than
a new type of analysis. Furthermore, they were apparently restricted
in application. Fermat knew how to make use of them only in the
case of rational expressions, whereas Newton and Leibniz, through
their application of infinite series, recognized the universality of such
procedures. Nevertheless, no mathematician, with the possible ex-
ception of Barrow, so nearly anticipated the invention of the calculus
as did Fermat.
The influence of Fermat on his contemporaries and immediate suc-
cessors 228 is difficult to determine. Probably his work was not so well
known as that of Cavalieri, for the latter was widely read in his two
famous books, whereas Fermat did not publish either his methods or
his results. For this failure to publish his work, Fermat, it has been
said, 229 lost the credit for the invention of the calculus; but such an
assertion is incorrect. In the first place, it is clear that he cannot be
thought of as its inventor. Secondly, his work was collected and printed
posthumously as the Opera varia, in 1679, before either of the earliest
published accounts of the calculus by Newton or Leibniz had appeared.
In spite of the fact that Fermat did not himself publish his methods,
227 Lagrange, Laplace, and Fourier have so called him, but Poisson has correctly pointed
out that Fermat does not deserve such a designation, inasmuch as he failed to recognize
the problem of quadratures as the inverse of that of tangents. The relevant statements
of these four men may all be found in Cajori, "Who Was the First Inventor of the Calcu-
lus?" Cf . also Marie, Histoire des sciences mathematiques, IV, 93 ff. Sloman has very unfairly
said that "Fermat hardly deserves to be named at all" in this connection. See his Claim of
Leibnitz to the Invention of the Differential Calculus, pp. 45-47.
228 See Genty, L 'Influence de Fermat sur son siicle. This book is more concerned, however,
to show the priority and independence of Fermat's results than specifically to point out
their effects on others.
229 Dantzig, Number, the Language of Science, pp. 131-32.
A Century of Anticipation 165
these became known through his correspondence with such men as
Roberval, Pascal, and Mersenne, as well as through the publication by-
others during his lifetime of portions of it. As a consequence, his work
was in large part responsible for a number of transition methods which
appeared just before the advent of the calculus. It is therefore scarcely
correct to say, with Lagrange, 230 that "Fermat's contemporaries did
not seize the spirit of this new type of calculus." Infinitesimal con-
siderations such as those of Fermat constituted a large portion of the
mathematical activity of the period. Nevertheless, there was indeed
one great mathematician who remained somewhat cool toward the
new views, even though in his early years he had made effective use
of them. This was Rene Descartes, the severest critic of Fermat.
The very first mathematical production of Descartes was an attempt,
in 1618, to deal with the laws of falling bodies by means of infinitesimals.
In this he made an error, in that he assumed, as Galileo had also in
1604, 281 that the velocity was proportional to the distance rather than
the time; but if the distance axis in his demonstration were changed to
the time axis, his procedure would be essentially that which Oresme
and other Scholastics had used. 232 Descartes was probably familiar
with their works and may have derived this form of proof, as well as
suggestions on analytic geometry, from reading Oresme. 233
At any rate, Descartes was acquainted with ancient, medieval, and
modern views on infinitesimals, and used them. In a second memoir of
about the same time he wrote on fluid pressure. In this connection he
may have known through Beekman of Stevin's work on infinitesimals.
At all events, in considering the force drawing a body, he used such
phrases as the "first instant of its movement," and the "first imaginable
speed." 234 Some years later, in 1632, he answered correctly a number of
00 See Brassine, PrScis des ceuvres math&matiques de P. Fermat, p. 4.
ni Duhem, fitudes sur Lionard de Vinci, III, 564.
m See Descartes, (Euvres, X, 219; cf. also X, 59, 76-77.
*» There is a great difference of opinion on this subject. Wallner ("Entwickelungsge-
schichtliche Momente," p. 120) sees not the least influence of Oresme on Descartes;
Stamm ("Tractatus de continuo," p. 24) says that the problem of the latitude of forms of
Oresme was undoubtedly the most important influence on Descartes; Wieleitner ("Ueber
den Funktionsbegriff," p. 242) says that Descartes undoubtedly knew of Oresme's work,
but that the essential idea of the dependence of quantities found in analytic geometry is
missing in Oresme; Duhem (Etudes sur Lionard de Vinci, HI, 386) has claimed that Oresme
created analytic geometry.
«* Milhaud, Descartes savant, pp. 162-63.
166 A Century of Anticipation
questions, which Mersennc had sent him, on areas, volumes, and
centers of gravity connected with the parabolas y n = px — problems
similar to those which Fermat had solved. Descartes did not tell what
method he had used, but it was probably a skillful application of the
methods of Archimedes, Kepler, and Cavalieri. 235 However, after the
publication of his famous Geometrie in 1637, Descartes' interest in
the subject began to wane, 236 for his mathematical work was only an
episode in the development of his philosophy. Consequently he did not
participate effectively in the development of the infinitesimal methods
which preoccupied the minds of most mathematicians at this time. 237
The acrimonius quarrel with Fermat, however, sustained his interest
in the problem of tangents and led him to considerations which, if
pursued further, might have been more effective than the infinitesimal
methods in leading to a clearer understanding of the basis of the cal-
culus.
The ancients, with the possible exception of Archimedes, had not
developed a general definition of a tangent to a curve nor any method
of determining it. Descartes, however, realized more fully than a
number of his contemporaries that this constituted not only "the
most useful and general problem that I know but even that I have
ever desired to know in geometry. " 238 He thereupon elaborated his
celebrated method of tangents in terms of the equality of roots.
Descartes' method consisted in passing through two points of the
curve a circle with its center on the x axis, and then making the points
of intersection coincide. The center of the circle thus becomes the
point on the x axis through which the normal to the curve passes, and
the tangent is consequently known. This may be illustrated in some-
what simplified form as follows: Let the tangent at the point (a, a) on
the parabola y 2 = ax be required. The equation of the circle going
though (a,a) and having its center on the axis is x 2 + y 2 — Ihx + 2ah
— 2a 2 = 0, where h is undetermined. Substituting ax for y 2 , the quan-
tity h may then be so determined that the resulting equation has equal
roots — that is, so that the intersections of the circle with the parabola
*" Ibid., pp. 164-68.
at Ibid., p. 246; cf. also Marie, Histoire des sciences mathimatiques et physiques, IV, 21.
m Milhaud (Descartes savant, pp. 162-63) denies that it was the insistence on clear ideas
which made Descartes avoid the use of infinitesimals,
w Descartes, (Euvres, VI, 413.
A Century of Anticipation 167
coincide. 239 This value, h = fa, is the abscissa of the point on the axis
through which the normal to the parabola passes. The tangent is then
the line through (a, a) perpendicular to this normal.
It is to be remarked that Descartes' method is purely algebraic, no
concepts of limits or infinitesimals being manifestly involved. However,
any attempt to interpret geometrically the significance of the case in
which the roots are equal, to explain what is meant by speaking of co-
incident points, or to define the tangent to a curve, would necessarily
lead to these conceptions. If Descartes in his geometry had thought in
terms of continuous variables rather than of a correspondence between
symbols which represented lines in a geometrical diagram, 240 he might
have been led to interpret his tangent method in terms of limits, and
so have given a different direction to the anticipations of the calculus.
His algebra, however, was still grounded in the geometry of lines, and
the idea of a continuously varying quantity was not really established
in analysis until the time of Euler. 241
In criticising Fermat's method of tangents, Descartes attempted to
correct the method by interpreting it in terms of equal roots and co-
incident points, a procedure which was practically equivalent to de-
fining the tangent as the limit of a secant. 242 Descartes did not express
himself in this manner, however, inasmuch as the concept of a limit
was far from clear at this time. Fermat, who was thinking of infinitesi-
mals, could not see that his method had anything in common with the
algebraic (limit) method of Descartes and so precipitated a quarrel as
to priority, one of the many which the seventeenth century produced
as the result of the confusion of thought as to the basis of infinitesimal
methods. Descartes preferred his method of tangents to that of Fermat
because of the apparent freedom from the concept of infinitesimals, al-
though its application was frequently much more tedious and was
limited to algebraic curves.
In finding the tangent to a nonalgebraic or "mechanical" curve, such
as the cycloid, Descartes in 1638 made use of the concept of an in-
stantaneous center of rotation. This is, of course, also directly con-
nected with the use of limits and infinitesimals, but is expressible
m Ibid., VI, 413-24; cf. also Voss, "Calcul differentiel," p. 244.
"• Descartes, (Euvres, VI, 369; cf. also VI, 411-12.
^ Fine, Number-System, p. 121.
"" See Duhamel, "M6moire," pp. 298-308; Milhaud, Descartes savant, pp. 159-62.
168 A Century of Anticipation
without the use of such terminology, through the circumlocution af-
forded by the notion of instantaneous velocity. Supposedly, this notion
is intuitively clear, but at the time it was not rigorously defined. It
had been made acceptable, moreover, by the work of Galileo, 243 and
Roberval and Torricelli were employing it in geometry at practically
the same time as Descartes. Descartes' reasoning was as follows: If a
polygon is rolled along a straight line, any vertex will describe a series
of circular arcs, the centers of which are the points on the line which
the vertices of the polygon touch: i. e., in rolling the polygon along the
line, we rotate the polygon about each of these points in turn. The
cycloid, now, is the curve generated by a point on a circle: i. e., by a
vertex of a polygon with an infinite number of sides, as it is rolled
along a line. The cycloid is therefore made up of an infinite number
of circular arcs, and the tangent at any point, P is therefore perpen-
dicular to the line joining P to the point Q in which the generating
circle touches the base line. Inasmuch as Q can easily be determined,
the tangent at P can be drawn. 244
In Descartes' work one sees an avoidance of the idea of an infinitely
small quantity in mathematics and the use, instead, of algebraic and
mechanical conceptions. Whereas Fermat saw only the practical ad-
vantages of the infinitesimal methods, Descartes saw better the risks
they entailed. Descartes' evasion of them was of course justified by the
lack of a clear theoretical basis for infinitesimal reasoning, but it was
opposed to the mathematical trend of the time. We have seen that in
the years following his Discours de la methode, there appeared in print
an unprecedentedly large number of works devoted to infinitesimal
methods. In most of them the work was largely based on synthetic
geometry, although Roberval and Pascal showed an arithmetizing
tendency in their quadratures. In France only Fermat made effective
use, in the anticipations of the calculus, of the new analytic methods
which he and Descartes were developing. In England, however, the
mathematician and theologian John Wallis applied analytic geometry
to the problem of quadratures with comparable success.
John Wallis had become familiar with analytical methods largely
241 De Giuli ("Galileo e Descartes") asserts that Descartes owed to Galileo much of his
philosophic method.
M Walker, A Study of the Traits des Indivisibles of Roberval, pp. 137-39.
A Century of Anticipation 169
through Harriot. In his work on conic sections, Wallis followed Viete,
Descartes, Fermat, and Harriot in the application of literal algebra
to the problems of geometry. Wallis, however, went far beyond these
men in that he sought to free arithmetic completely from geometric
representation, a goal which he thought would be easily reached. 246 He
first showed how all the theorems of Euclid V could be derived arith-
metically without difficulty, and then in algebra broke away from the
idea, derived from geometry, that the terms of an equation must be
homogeneous. Luckily, Wallis did not worry overmuch about mathe-
matical rigor: we know now how difficult the arithmetizing of mathe-
matics was to be.
Instead of observing the caution which classical ideas of rigor ex-
acted, Wallis was influenced by prevailing thought to make free use of
analogy and incomplete induction in his work, as well as of the con-
cepts of infinity and infinitesimals, which had not yet been rigorously
established. We have seen these tendencies developing in the work of
Cavalieri and Fermat and, in continuing this tradition, Wallis came
nearer to the limit concept than did any other of Newton's predeces-
sors. It is clear that this notion is implicit in the work of most of his
French and Italian contemporaries, but it was not expressed by them.
Instead, the concept of the infinitesimal was employed. With Wallis'
arithmetical point of view, however, one is brought again into more
direct touch with the limit idea which the mathematicians of the Low
Countries — Stevin, Gregory of St. Vincent, and Tacquet — had sought
to formulate.
To what extent Wallis was influenced by the ideas of these other
men is difficult to determine. He wrote the most complete treatise on
statics since the time of Stevin, 246 and may well have been familiar
with the arithmetical limit methods of the latter, either directly or
through the somewhat similar work of Roberval. 247 Wallis admitted
that upon the advice of Wren he read, about 1652, part of the Opus
geometricum of Gregory of St. Vincent, but he added that in this he
did not run across any propositions new to him. 248 On the other hand,
Wallis and Tacquet were probably independent of each other in their
245 Prag, "John Wallis." *• Duhem, Les Origines de la statique, II, 211.
247 Walker, A Study of the TraiU des Indivisibles of Roberval, p. 77; cf. also p. 165.
248 Wallis, Opera mathematico, Vol. II, Arithmetica infinitorum, Preface.
170 A Century of Anticipation
work, inasmuch as their books on this subject appeared almost simul-
taneously — in 1655 and 1656 respectively. Oddly enough, moreover,
although Wallis displays the arithmetizing and limit tendencies of
Tacquet, the chief inspiration for his work came rather from reading,
in 1650, the geometrical method of indivisibles of Cavalieri as ex-
pounded by Torricelli. This he professed in the prefaces of two of his
books: De sectionibus conicis tractatus, and Arithmetica infinitorum sive
nova methodus inquirendi in curvilineorum quadraturam. 249 Cavalieri's
work, however, had been almost purely geometrical, whereas Wallis
proceeded largely arithmetically, and in the end abstracted from the
geometry of indivisibles the arithmetic notion of a limit. "Following
Oughtred, Descartes, and Harriot," he applied to his demonstrations
the symbolism of arithmetic in order to give to them "at the same time
the maximum brevity and perspicuity." The use of arithmetic calcu-
lation he held to be simpler and not less "legitimate or scientific" than
that by lines. 250
The manner in which Wallis made the transition from the geometry
of lines to the arithmetic of numbers is brought out clearly in his proof
that the area of a triangle is the product of the base by half the alti-
tude. 261 He assumed at the outset, as had Cavalieri, that a plane figure
may be regarded as made up of an infinite number of parallel lines —
or rather, as he preferred, of an infinite number of parallelograms, the
altitudes of which are equal, that of any one of them being — , or an
infinitely small aliquot part of the altitude of the figure. 252 Here we
have not only the first appearance of the symbol o° for infinity, 253 but
also the earliest use of the Scholastic categorematic infinity in the
field of arithmetic. Furthermore, the treatment by Wallis of the in-
finitely small is far more daring and decisive than that of Fermat.
Whereas the latter had not expressly called his symbol E an infinitesi-
mal, Wallis for his part said that _L represented an infinitely small
oo
quantity, or non-quanta. A parallelogram whose altitude is infinitely
*• Opera motketnatico, Vol. II.
260 Opera mathematica, Vol. II, De sectionibus conicis, "Dedicatio," and also p. 3.
851 Ibid., pp. 4-9. A good summary of the work of Wallis along these lines is to be found
in Sloman, The Claim of Leibnitz to the Invention of the Differential Calculus, pp. 8 ff .
** Opera mathematica, Vol. II, De sectionibus conicis, p. 4.
m "Esto enim oo nota numeri infiniti." Ibid.
A Century of Anticipation 171
small or zero is therefore "scarcely anything but a line," except that
this line is supposed "extensible, or to have such a small thickness that
by an infinite multiplication a certain altitude or width can be ac-
quired." 264
Returning to the proposition on the area of the triangle, Wallis sup-
posed this to be divided into an infinite number of lines, or infinitesimal
parallelograms, parallel to the base. The areas of these, taken from the
vertex to the base, form an arithmetic progression beginning with zero.
Moreover, there is a well-known rule that the sum of all the terms in
such a progression is the product of the last term by half the number
of terms. Since "there is no cause for discrimination between finite and
infinite numbers," it can be applied to the areas in the triangle. If the
altitude and base of the triangle are taken as A and B respectively, the
area of the last parallelogram in the progression will then be — A3.
1 °°
The area of the whole triangle is therefore J— A 3. 9 , or \A3. m He
00 *
then applied a similar type of argument to numerous quadratures and
cubatures involving cylinders, cones, and conic sections.
Wallis realized that his procedure was highly unorthodox, but he
said that it could be verified by "that very well-known apagogic
method" of in-and-circumscribed figures. To give this Wallis felt
would be superfluous, because "the frequent iteration would create
nausea in the reader." Furthermore, he said that anyone versed in
mathematics could supply such proof, since it occurred frequently
among the ancients and the moderns. 256 Modern mathematics has
found it necessary to modify greatly the view of the infinite which
Wallis held and to banish entirely his infinitely small magnitudes.
Nevertheless, the development of the calculus is the result of efforts,
such as his, to substitute for the prolixity of the method of exhaustion
a direct arithmetical analysis.
The procedure of Wallis in De sectionibus conicis was based largely
upon crude manipulations of his symbol o° . In the Arithmetica in-
finitorum, however, he pursued similar investigations from a somewhat
different point of view — one resembling more closely the arithmetic
*"Ibid.
156 Ibid., pp. 8-9; cf. also Opera maihematico, Vol. II, Arithmetica infinitorutn, p. 2.
*• Opera mathematica, Vol. II, De sectionibus conicis, p. 6.
172 A Century of Anticipation
methods of Stevin and Roberval and the limit concept. In this con-
nection he demonstrated the equivalent of the theorem : JqX dx = ,
« + 1
apparently unaware that this proposition had appeared in numerous
forms during the preceding twenty years. Wallis reached this conclusion
by observing first the equalities
+ 1 = ,. + 1 + 2 = , + 1+2 + 3 = le
1 + 1 2 '2+2 + 2 2 '3 + 3 + 3 + 3 "
In these the ratio is \ for any finite number of terms, from which Wallis
concluded that this will be the ratio likewise for an infinite number.
Through this, Wallis arrived at an alternative form of demonstration
for the theorem above on the area of a triangle. 257
In proceeding further, Wallis noted that in the equalities
+ 1 1 . + 1+4 _ 1 , ,, + 1+4 + 9 , , .
nri-' + *>4 + 4 + 4" 5+ ™'9 + 9 + 9 + 9~ 5 + Tlr '---'
the greater the number of terms, the more closely does the ratio ap-
proximate to |, so that "at length it differs from it by less than any
assignable magnitude." If this is continued to infinity, the difference
"will be about to vanish completely." Consequently, the ratio for an
infinite number of terms is i. 268
Wallis then proceded to observe, in a similar manner, that the
analogous ratios for the third, fourth, fifth, and higher powers of the
integers are respectively \, i, i, and so on. He then affirmed the validity
of the rule for all powers, rational or irrational (except, of course,
— I). 269 This extension of the work of his French and Italian prede-
cessors was made upon the basis of what Wallis spoke of as interpola-
tion and induction. By the former, he seems to have had in mind a
principle of continuity, 260 or of permanence of form, by which the rule
could be asserted to hold for values intermediate between those for
which it was known to be valid. By the latter, Wallis meant not mathe-
matical or complete induction but induction in the scientific sense,
267 Opera mathematica, Vol. II, Arithmetica infinitorum, pp. 1-3; cf. also p. 157.
™Ibid., pp. 15-16; cf. also p. 158.
•*• Ibid., pp. 31-53. Wolf 04 History of Science, Philosophy, and Technology in the Six-
teenth and Seventeenth Centuries, p. 209) has incorrectly stated in this connection that the
predecessors of Wallis confined themselves to positive integral powers. Furthermore, he
neglects to state that Wallis extended the rule to irrational powers also,
»° Nunn, "The Arithmetic of Infinities."
A Century of Anticipation 173
analogy similar to that by which we have seen him conclude properties
of the infinite from those of the finite. In this respect his work is a
good indication of the looseness of thought at the time. More signifi-
cantly, his extension of the rule to irrational powers indicates a tend-
ency to break away from the persistent idea, derived from Pythag-
orean geometry, that irrational magnitudes are not numbers in the
strict sense of the word. This was in line with his declaration of the
independence of arithmetic from geometry, a freedom which was neces-
sary for the later elaboration of the limit concept which he was here
adumbrating.
The proposition above, on the ratios of the powers of integers, Wallis
then applied to problems on quadratures and cubatures. In this re-
spect he may be said to have determined the areas and volumes as
limits of infinite sequences, in much the same way as Fermat had found
them by means of infinite geometric progressions. In fact, the basis for
the concept of the definite integral may be considered fairly well estab-
lished in the work of Fermat and Wallis, although it was to become con-
fused later by the introduction of the conceptions of fluxions and differ-
entials. However, that neither of these men realized fully the significance
of this concept is seen in their lack of clarity. We have seen that Fermat
did not fully explain the nature of his symbol E. Wallis confused his
work with the infinitesimal, identifying infinitely small rectangles with
lines, and writing J_ = — ideas which were to lead to the conception,
oo
found in Leibniz, of the integral as a sort of totality, rather than as the
limit of a sum.
Wallis was interested in another question which characteristically
concerned mathematicians of the time — that of the angle of contact
(horn angle) formed by two curves with a common tangent, a figure
which Euclid, Jordanus Nemorarius, Cardan, and many others had
considered. Discussion on this point may have been instrumental in
maintaining the concept of the infinitely small as a valid notion, for
it lent plausibility to the idea of an ultimate indivisible, smaller than
any assignable magnitude and yet seemingly not the same as absolute
zero. 261 The postulate of Archimedes of course excluded such angles
261 For a full discussion of the history of this subject, see Vivanti, 77 concetto, or the French
summary of this book given in Bibliothica Mathematica, N. S., VIII (1894) as "Note sur
l'histoire de Pinriniment petit."
174 A Century of Anticipation
as magnitudes, just as it had excluded other infinitesimals; but mathe-
maticians of the seventeenth century regarded them as interesting il-
lustrations of their concepts, and discussed the question as to whether
they were zero or not. Galileo, Wallis, 262 and others asserted that such
angles were absolutely zero, whereas Hobbes, Leibniz, and Newton,
for example, held that they were in some way different from zero. On
this point argument was possible, of course, only for two reasons: the
general lack of critical definitions during this period (a rigorous defi-
nition of the tangent to a curve, although implied by the work of
Descartes, Fermat, and others, having not been given at the time), and
the failure to distinguish clearly between the geometrical figure and
its arithmetical measurement. Both of these shortcomings were to be
significant later in exposing the calculus of Newton and Leibniz to
severe criticism.
After a century of doubt, clear definitions were formulated and the
calculus was established upon arithmetical rather than geometrical
conceptions. The work of Wallis was an attempt to bring about such
an arithmetization, and in this respect it won the support of his con-
temporary, James Gregory. The latter, in his Vera circuli et hyperbolae
quadratura of 1667, viewed the passage to the limit as an independent
arithmetical operation, suitable as a means of defining new numbers
not belonging to the ordinary irrationals. 263 In connection with this
work, he constructed in and circumscribed polygons to the circle and
hyperbola and showed that, by doubling the number of sides of these,
converging series were obtained in which the difference became smaller
and smaller. These series consequently had a limit, which, "if one may
speak in this manner," could be considered the last polygon in each
series. This consequently would give the area of the curvilinear figure. 264
The areas of the circle and the hyperbola Gregory gave to as many as
twenty-six figures, although the limit he recognized as in general in-
commensurable. 266
This work on the limit of converging infinite series represented a
generalization of the earlier propositions of Gregory of St. Vincent and
Tacquet on geometric progressions, with which James Gregory was
M * Opera mathematica, Vol. II, De angulo contactus et semicirculi.
*«* Wallner, "Uber die Entstehung des Grenzbegriffes," p. 258; cf. also Georg Heinrich,
"James Gregorys 'Vera circuli et hyperbolae quadratura.' "
*•* Vera circuli et hyperbolae quadratura, pp. 15-16. 2M Ibid., pp. 48, 25.
A Century of Anticipation 175
acquainted. 266 He may also have been familiar with the somewhat
similar arithmetic work of Roberval, for in another connection he
used geometric transformations resembling the so-called Robervallian
lines. 267 However, whereas Roberval and Wallis had been led to their
arithmetization through the method of indivisibles, Gregory preferred
to employ in quadratures the indirect method of the ancients, showing
that the difference can be made less than any given quantity. 268
Nevertheless, in connection with this work he adopted the newer ana-
lytic methods of Descartes. In this respect also he followed the method
of Fermat for determining tangents to curves. For example, the tangent
to y* = x 2 (a + x) he found at the point for which x = b as follows: 269
Choose a second point, with abscissa less than a; by a vanishing small
amount, o, 270 and assume, "if we may do so," that the corresponding
ordinate may be taken indifferently as that of the curve or of the tan-
gent. Then set up the suitable proportion, divide by o, and reject the
terms in which o or a power of it remain. The subtangent is in this way
3b 2 + 3ab
found as z = .
3b + la
Although Gregory did not refer to Fermat in this connection, it is
obvious that the methods of the two men are identical with the ex-
ception that the E of Fermat has been changed to o, a change of no-
tation which was to be adopted a year or two later by Newton, perhaps
under the influence of Gregory's work. The type of arithmetical and
analytical work of Fermat, Wallis, and Gregory represented the tend-
ency which was to lead to the calculus, but it met with almost im-
mediate opposition, for the spirit of the age was directed toward the
solution of problems through geometrical considerations. Such an
arithmetization of mathematics was opposed with particular vigor by
two Englishmen, the philosopher Thomas Hobbes and the mathe-
matician and theologian Isaac Barrow. 271 Hobbes objected strenuously
to "the whole herd of them who apply their algebra to geometry." 272
»• Ibid., p. 20; cf. also p. 123. M7 Galloys, "Reponse a l'Scrit de M. David Gregorie."
468 See Geometriae pars universalis, pp. 27-29, 74 ff.
»• Ibid., pp. 20-22. » "Nihil seu serum o." Ibid.
m See Cajori, "Controversies on Mathematics between Wallis, Hobbes*, and Barrow."
A summary of this article is given in Bulletin, American Mathematical Society, XXXV
(1929), 13.
272 Weinrich, Ober die Bedeutung des Hobbes fur das naturwissenschafiliche und mathe-
matiscke Denken, p. 91.
176 A Century of Anticipation
He maintained that they mistook the study of symbols for that of
geometry, and characterized the Arithmetica infinitorum as a "scurvy
book." 273 He referred to the arithmetization represented by Wallis as
absurd and as "a scab of symbols." 274
This attitude toward algebra and analytic geometry was probably
the result not only of the general predilection in the seventeenth
century for geometrical rather than arithmetical methods, but also of
Hobbes' exaggerated view of mathematics as an idealization of sensory
perception, rather than as a branch of abstract formal logic. Greek
thought had accepted mathematics as derived from the experience of
the senses by the abstraction from concrete objects of irrelevant
properties. Hobbes, however, was unwilling to regard lines as deprived
of all breadth, or surfaces of all thickness. 276 Consequently, the in-
finitely small was for him merely the smallest possible line, plane, or
solid — a view of infinitesimals held by the school of mathematical
atomists in antiquity and not unlike that of Cavalieri.
Hobbes' view of number was comparable to his attitude toward the
geometrical elements. He adopted the Pythagorean notion of number
as a collection of units, and he interpreted ratios only in terms of
geometrical considerations. 276 This attitude would not only operate
against the free use of arithmetic processes which Wallis carried out on
the basis of analogy or induction, but would, in fact, when combined
with Hobbes' naive view of geometrical magnitude, make the dis-
tinction between rational and irrational number, and the consequent
introduction of the limit concept, logically superfluous. Hobbes' ideas
in this direction were, as a result, of little consequence in the develop-
ment of the concepts of the calculus. The.e is, however, another aspect
of his thought which may have been more significant.
We have mentioned that Aristotle thought of motion as the fulfill-
ment of the potential, thus centering attention not on the mathematical
aspect of motion, but rather on the metaphysical, attributing to a body
in motion a striving toward a goal. It could not of itself, however, reach
its goal, since, in order that it do so, the constant application of a force
was necessary. In the fourteenth century this Peripatetic theory was
questioned, because it could not "save the phenomena." In its place,
« English Works, VII, 283. K4 Cf. ibid., VII, 187, 361 ff.
» Ibid., VII, 67, 200 ff., 438. m Cf. Opera omnia, IV, 27, 36.
A Century of Anticipation 177
Jean Buridan had substituted the doctrine of impetus (or inertia), or
the tendency of a body to remain in motion. 277
This Scholastic view of impetus gave to motion a so-called intensive
characteristic, for it centered attention upon the act of moving, rather
than on change of position or extension. Such a shift of emphasis made
acceptable the notion of motion at a point, an idea which Aristotle
had specifically rejected. 278 This was followed immediately by the
quantitative treatment of instantaneous velocity found in Calculator,
Hentisbery, and Oresme. The idea of impetus was familiar to Nicholas
of Cusa, Leonardo da Vinci, and others of the fifteenth and sixteenth
centuries, although it was often associated with, and obfuscated by,
accretions of Neoplatonic mysticism and of vitalistic and teleological
thought, such as are to be found in the works of Nicholas of Cusa,
Paracelsus, and Kepler. 279 In 1638 the idea of impetus or inertia cul-
minated in the famous clarification of the laws of dynamics given by
Galileo; and at the same time the concept of instantaneous velocity
was likewise successfully applied in geometry by Torricelli, Roberval,
and Descartes.
It must be kept in mind, of course, that no definition of instanta-
neous velocity had at the time been given, nor could it have been
framed before the development of the differential calculus. However,
the concept of inertia was to make the idea of motion at a point in-
tuitively and scientifically acceptable — as well as philosophically in-
teresting and geometrically useful — until such time as it could be made
mathematically rigorous.
Philosophers have generally displayed an interest in the calculus
because of its associations with problems of motion and variation which
present many intriguing metaphysical aspects. Hobbes was especially
concerned, inasmuch as he wished to make motion basic in his philo-
sophical scheme. His views were mathematically too naive to allow
of his adding to the discoveries being made in the new analysis, al-
though he may have had a somewhat broader influence on the later
m Duhem, Etudes sur Leonard de Vinci, III, vii-viii. m Physica VI. 234a.
2,9 The great German astronomer, for example, in his Mysterium cosmographicum and
his Astronomia nova, ascribed a vital faculty to the sun and the planets. In the 1621
edition of the former work, however, he said that one can substitute the word force for
soul. See Opera omnia, I, 174; II, 270; III, 176, 178-79, 313. See also Lasswitz, Geschichte
der Atomistik, II. 9-12; Duhem, Etudes sur Leonard de Vinci, II, 199-223.
178 A Century of Anticipation
thought and interpretation of the calculus. Hobbes was much im-
pressed with the success of Galileo's conception of the laws of motion,
in terms of inertia and changing velocity. Realizing that physics owes
its development to mathematical representation, 280 he wished to de-
scribe and explain these ideas in geometrical and metaphysical terms.
To this end, he introduced the concept of the conatus, 2 * 1 a beginning of
motion, analogous to the concept of the point as the beginning of
geometrical extension. Hobbes thus tried to emphasize, as had the
men of the fourteenth century, the idea of motion at a point, rather
than that of change of position. Not realizing, as Aristotle had dis-
cerned nineteen hundred years before, that instantaneous motion is
an intellectual — not an empirical — concept, he attempted to formulate
a definition in terms of his ingenuous nominalism, speaking of it as the
motion in an infinitely small interval — an interval less than any given
interval — that is, through a point. 282
Hobbes was not so fortunate, as Newton and Leibniz were later, in
formulating mathematically his concept of conatus. He did not under-
stand the relation between number and spatial quantity, nor did he
appreciate that instantaneous velocity is a purely numerical notion.
His views, however, were perhaps significant in their influence upon
the inventors of the calculus. The excessive nominalism of Hobbes was
to lead mathematicians away from a purely abstract view of the con-
cepts of mathematics, such as Wallis had displayed, and to induce them
to seek, during more than a century, for an intuitively, rather than a
logically, satisfactory basis for the calculus. It was largely on this ac-
count that both Newton and Leibniz sought to explain the new an-
alysis in terms of the percipient notion of the generation of magnitude,
rather than in terms of the logical conception of number only. 283 This
idea of generation is more immediately apparent in the method of
fluxions of the English empiricist, Newton ; but the German philoso-
pher, Leibniz, justified his differential method also in terms of the
analogous idea of continuity. Whereas Newton used the physical idea
of the "moment" of growing magnitudes, there grew up in Germany
a more metaphysical form of this in the notion of intensive magnitude
as opposed to extensive quantity. Upon this idea of a "tendency" or a
280 Opera omnia, II, 137. » Opera omnia, I, 177. «* Ibid.
283 Cf. Vivanti, // concetto d'infinitesimo, pp. 31-32.
A Century of Anticipation 179
"becoming," the mathematical infinitesimal throve, with the result
that philosophers have been reluctant to abandon it, even though
modern mathematics has shown that the basis of the calculus is to be
found in the derivative rather than in the differential.
There is, furthermore, an inclination on the part of philosophers (and
occasionally of mathematicians) to regard Hobbes' conatus, or the de-
rived concept of the intensive in motion, as the answer to Zeno's
paradoxes, inasmuch as even when the time interval has disappeared,
the tendency toward motion remains. 284 This attitude adds nothing to
the explanation of the paradoxes, for it fails to recognize that the con-
ception of motion at a point, which is the crux of the situation, is not
a scientific notion but a mathematical abstraction. As such, the logical
difficulties involved have all been cleared up by the calculus and the
mathematical continuum. Of course, there is nothing lost (or gained)
by calling the derivative (which is, after all, only a number) the in-
tensity at the point, for this is only a change of name; but the answer
to the paradoxes must remain that given by mathematics, rather than
any made in an attempt to satisfy intuition.
Wallis' arithmetization (as well as Descartes' analytic geometry)
was criticized also by the contemporary mathematician, Isaac Barrow.
The latter wished to return to the Euclidean view, and maintained that
mathematical number has no real existence proper to itself and inde-
pendent of continuous geometrical quantity. Numbers like ^3, he
felt, cannot, even in thought itself, be abstracted from all magnitude.
Such "surd" numbers are "inexplicable," and, "having no merit of
their own, they are wont to be banished from arithmetic to another
science (which yet is no science), viz., algebra." Barrow held that
arithmetic is to be included in geometry, but that algebra is to be in-
cluded in logic rather than mathematics. 285
This view would, of course, lead directly away from the limit con-
cept which requires, for its effective use, as well as its logical defini-
tion, a conception of number not based on the geometrical interpre-
tation of continuous magnitude. The fact that Barrow advocated a
return to the classic conception of number and geometry may have in-
184 See Lasswitz, Geschichte der Atomistik, II, 30; Vivanti, II concetto d'infinilesimo, pp.
31-32.
m See Barrow, Mathematical Works, pp. 39, 45-46, 51-53, 56, 59.
180 A Century of Anticipation
fluenced his student, Newton, to seek to establish the calculus on the
idea of continuous variation found in motion and geometry, and to
avoid as far as possible the arithmetical notion of limit. Barrow's dis-
trust of algebraic methods may well have been responsible also for the
fact that he did not develop his geometrical discoveries into an effective
analytical tool, but in this respect he was not followed by Newton.
Although Barrow did not accept readily the algebra which Italian
mathematics had developed, he was impressed by the possibilities
offered by the conception of motion at a point, which had been used
to such advantage in geometry by Torricelli and others. Time he re-
garded as a mathematical quantity measurable by, although not de-
pendent on, motion, 286 and upon the suggestion of sensory evidence he
thought of it as a continuous magnitude, "passing with a steady flow."
This brought Barrow to the problem of the nature of the continuum
and the definition of instantaneous velocity. In his treatment of these
there is a complete lack of the limit concept and, instead, an attempted
blending of atomistic and kinematic views which may have been in
part the result of the influence of the Cambridge Platonists. "To every
instant of time, or indefinitely small particle of time, (I say instant or
indefinite particle, for it makes no difference whether we suppose a
line to be composed of points or of indefinitely small linelets; and so in
the same manner, whether we suppose time to be made up of instants
or indefinitely minute timelets) ; to every instant of time, I say, there
corresponds some degree of velocity, which the moving body is con-
sidered to possess at the instant." 287 This passage shows clearly that
Barrow's views are essentially infinitesimal and not much clearer than
those of Plato, Oresme, Galileo, or Hobbes. In fact, in his argument
that the area under a velocity-time curve represents distance, his
thought closely parallels that implied by Oresme and expressed by
Galileo. "If through all points of a line representing time are drawn. . .
parallel lines, the plane surface that results as the aggregate of the
parallel straight lines, when each represents the degree of velocity
corresponding to the point through which it is drawn, exactly corre-
sponds to the aggregate of the degrees of velocity, and thus most con-
veniently can be adapted to represent the space traversed also." 288
"• Gunn, The Problem of Time, p. 57.
» Barrow, Geometrical Lectures, p. 38. ** Geometrical Lectures, p. 39.
A Century of Anticipation 181
Barrow admitted that it may be contended rightly that very narrow
rectangles should be substituted for the lines, but maintained that "it
comes to the same thing whichever way you take it." 289
In pointing out that "time has many analogies with a line," Barrow
again suggested atomistic conceptions, saying that these magnitudes
could be considered as constituted either from the continuous flow of
one instant or point, or as an aggregation of instants or points. 290 Of
all the ways in which a continuous magnitude may be generated,
Barrow believed, with Cavalieri, that the one in which it is regarded
as composed of indivisibles, "in most cases is perhaps the most expe-
ditious of all, and not the least certain and infallible of the whole
set." 291 We have seen that Tacquet had attacked the method of in-
divisibles, substituting for it a sort of limit of infinitely small quan-
tities, and that Pascal followed him in this. Barrow, on the other hand,
defended Cavalieri 's method against the valid criticisms of Tacquet. 292
In spite of the lack of clarity and precision in his views on the con-
tinuum, the geometrical results of Barrow represent a remarkably
close approach to those of the calculus. They include not only nu-
merous theorems on quadratures and tangents, but also perhaps the
clearest recognition up to that time of the significance of the relation-
ship between these two types of problems. 293 All of his propositions,
however, are cast in geometrical forms which involve intricate and
unnatural constructions, instead of in the analytical symbolism of
Descartes, Fermat, and Wallis. If they were recast in terms of the
calculus, they would be equivalent to many of the standard rules and
theorems on differentiation and integration, including the fundamental
theorem of the calculus. 294 Any such attempt to interpret them in terms
of the present analytical notations, however, would be misleading. It
would suggest, by implication, the possession on the part of Barrow
of concepts equivalent to those of the derivative and the integral.
It would furthermore give to his work an analytic character which
the original was far from exhibiting.
With respect to the concepts involved in Barrow's geometry, we
have seen that his views indicate a return to the vague indivisible of
*»Ibid. «° Ibid., p. 37. ™ Ibid., p. 43. «• Ibid., pp. 44-46.
"• See Child, "Barrow, Newton and Leibniz, in Their Relation to the Discovery of the
Calculus."
** See Geometrical Lectures, pp. 30-32.
182 A Century of Anticipation
Cavalieri, rather than progress toward the development of the limit
concept fostered by Wallis. As to the form of his work, it is clear that
Barrow himself failed to appreciate it as a new type of analysis, suffi-
ciently significant in itself to be developed into an algorithmic pro-
cedure. He seems to have realized that he was indicating a new method
for finding tangents and areas, but by presenting it in synthetic form
he gave it the appearance of being an expansion of the classical ge-
ometry of the ancients. "These matters seem not only to be somewhat
difficult compared with other parts of Geometry, but also they have
not been as yet wholly taken up and exhaustively treated (as the other
parts have)." 295
There is one point in Barrow's Geometrical Lectures, nevertheless, at
which there is an indication that he may have used an analytic method
to arrive at his results, recasting these later into the synthetic form in
which he presented them. Also, in this connection, he constructed a
diagram which was to become significant in the calculus of Leibniz as
the familiar differential triangle. However, similar figures had ap-
peared in the geometry of Torricelli, Roberval, Pascal, and Fermat.
At the close of Lecture X, Barrow said:
We have now finished in some fashion the first part, as we declared, of
our subject. Supplementary to this we add, in the form of appendices, a
method for finding tangents by calculation frequently used by us. Although
I hardly know, after so many well-known and well-worn methods of the
kind above, whether there is any advantage in doing so. Yet I do so on the
advice of a friend [later shown to be Newton 296 ]: and all the more willingly,
because it seems to be more profitable and general than those which I have
discussed. 297
If Barrow had believed that he was inventing a new subject, it seems
likely that it is this method which he would have put forward, instead
of his classical proofs.
Let AP % PM be two straight lines given in position, of which PM cuts
a given curve in M , and let MT be supposed to touch the curve at M , and
to cut the straight line at T [fig. 22].
In order to find the quantity of the straight line PT,I set off an indefi-
nitely small arc, MN, of the curve; then I draw NQ, NR parallel to MP,
AP; I call MP = m, PT = t, MR = a, NR = e, and other straight lines,
determined by the special nature of the curve, useful for the matter in
"• Geometrical Lectures, p. 66.
"• See More, Isaac Newton, p. 185, n. m Geometrical Lectures, p. 119.
A Century of Anticipation
183
hand, I also designate by name; also I compare MR, NR (and through
them, MP, PT) with one another by means of an equation obtained by
calculation; meantime observing the following rules.
Rule 1. In the calculation, I omit all terms containing a power of a or e,
or products of these (for these terms have no value).
Rule 2. After the equation has been formed, I reject all terms consisting
of letters denoting known or determined quantities, or terms which do
not contain a or e (for these terms, brought over to one side of the equation,
will always be equal to zero).
Rule 3. 1 substitute m (or MP)' for a, and t (or PT) for e. Hence at length
the quantity of PT is found. 298
From this passage one observes that the "method for finding tan-
gents by calculation" used by Barrow resembles very closely the pro-
T Q
FIGURE 22
cedure now employed in the differential calculus, the letters a and e
being equivalent to the customary symbols Ay and Ax. This is an
elaboration of Fermat's method, in which, however, only one infinitely
small quantity, E, had been used. Barrow's method constitutes an
improvement upon that of Fermat in that it makes more convenient
the application of the method to implicit functions. Barrow apparently
did not know directly of Fermat's work, for he nowhere mentioned
his name. Nevertheless, the men to whom he referred as the sources
of his ideas include Descartes, Huygens, Galileo, Cavalieri, Gregory of
St. Vincent, James Gregory, and Wallis, 299 and it is possible that
Fermat's method became known to Barrow through them. Huygens
and James Gregory, in particular, made frequent use of Fermat's
«• Geometrical Lectures, p. 120. "• See Geometrical Lectures, p. 13.
-1
184 A Century of Anticipation
characteristic procedure; and Newton, at any rate, recognized that
Barrow's rule was but an improvement upon this method of drawing
tangents. 300
Although the tangent method of Barrow resembles the process of
differentiation even more closely than does that of Fermat, one is not
at liberty to impute to him the concepts implied by our symbols Ay
and Ax. He was quite evidently thinking in terms of geometrical
problems and infinitesimals, rather than of functions and symbols for
continuous variables. His statement that a and e "have no value" is
equivalent in this respect to Fermat 's neglect, at the end of the cal-
culation, of all terms involving E. Neither Barrow nor Fermat justi-
fied the neglect of these terms, for neither one had a clear conception
of a limit. Fermat, perhaps, came nearer to such an idea in thinking
of his pseudo-equality as a rigorous equality only when E was zero.
Barrow did not tell why the higher powers are to be neglected. The
first and third rules given by Barrow in his method are, of course,
logically to be justified only in terms of limits. Barrow undoubtedly
thought of the triangle MRN as rigorously congruent to MTP only
when it was infinitesimally small, for he said here and elsewhere, that
"if the arc MN is assumed to be indefinitely small, we may safely sub-
stitute instead of it the small bit of the tangent." 301 Passages such as
these reappear in the work both of Newton and Leibniz, showing how
difficult it was at that time for mathematicians to think arithmetically
in terms of limits.
Of all the mathematicians who anticipated portions of the differ-
ential and the integral calculus, none approached more closely to the
new analysis than did Fermat and Barrow. The former invented
analytic methods of procedure equivalent both to differentiations and
to integrations, ,but he appears not to have realized fully the signifi-
cance of the interrelation between the two types. On the other hand,
Barrow appears to have discovered the fundamental inverse relation-
ship, but because he did not develop fully the possibilities of the ana-
lytic representation of the operations involved, he was unable to make
effective use of it. 302 He systematically reduced inverse-tangent prob-
300 More, Isaac Newton, p. 185, n. 301 Geometrical Lectures, p. 61 ; cf. also pp. 120-21.
302 See Zeuthen, "Notes," 1897, pp. 565-606; or his Gesckichte der Malhematik itn XVI.
una 1 XVII. Jahrhundert, pp. 345-57, for excellent analyses of this portion of Barrow's
work.
A Century of Anticipation 185
lems to quadratures, but he did not convert the latter, by means of his
inversion theorem, into considerations derived from tangent deter-
minations — that is, he did not express them in terms of the anti-
derivative, as is generally done in the calculus. Barrow saw no ad-
vantage in doing this, because he had not reduced his tangent method
to a simple algorithmic form, as did Newton and Leibniz shortly
thereafter. Had he done so, he would without doubt have forestalled
these men as the founder of the calculus. 303
This, however, Barrow did not even attempt to do. After preparing
his lectures for publication, he turned them over to Newton and Collins
for final revision, and gave up mathematics for the study of theology.
Newton displayed the analytical knowledge required to develop the
geometrical ideas of Barrow into an algorithm, and in fact was already
in possession of his methods of the calculus several years before 1670,
the year in which Barrow's Geometrical Lectures were published. In
fact, there had appeared earlier in the Low Countries a number of rules
which in application so closely resemble parts of the calculus that they
may be thought of as forming a transition from the infinitesimal pro-
cedures, developed during this century of anticipation, to the methods
of fluxions and differentials. Such rules for tangents and for maxima
and minima as those formulated by Sluse, Hudde, and Huygens did
not involve new basic conceptions. They were simply canonical forms
derived from earlier methods, particularly those of Fermat and Barrow.
Sluse, for example, was perhaps the first mathematician to give a
general algorithm for writing down, without following through the
analytic calculation required by such methods as those of Barrow and
Fermat, the tangent to a curve whose equation is rational in x and
y. 30i His rule, which he seems to have formulated in 1652 306 but which
was not published until 1673, may be stated in the following terms. Let
the equation be f(x } y) = 0. Then the sub tangent will be the quotient
obtained by placing in the numerator all the terms containing y, each
multiplied by the exponent of the power of y appearing in it; and in the
denominator all the terms containing x, each multiplied by the ex-
** Child would categorically attribute the invention to Barrow, because, using modern
analytical notations instead of the synthetic form of the original, he reads into this work
algebraic methods of procedure, rather than geometric proofs of propositions.
"* Rosenfeld, "Ren6 Francois de Sluse et le probleme des tangentes."
** Le Paige, "Correspondance de Rene" Francois de Sluse."
186 A Century of Anticipation
ponent of the power of x appearing in it and then divided by x. m
. v f (x y)
This is, of course, equivalent to forming the quotient ' y ' , but
/»(*, y)
Sluse was not thinking in terms of derivatives of functions, as we do.
He did not give a general demonstration of his rule, but such is easily
supplied by the methods of either Fermat or Barrow.
Johann Hudde, in 1659, gave an exactly analogous rule, 307 and this
was rediscovered a few years later by Christiaan Huygens. 308 Hudde
stated also a rule for writing down, without carrying out the work of
Fermat's method, the maximum or minimum value of a rational func-
tion of one variable. 309 This is equivalent to setting equal to zero the
derivative of the function as found by the rule for quotients given in
calculus textbooks.
The almost simultaneous appearance of such rules and formulas in-
dicates that shortly after the middle of the seventeenth century in-
finitesimal considerations were so widely employed and had developed
to such a point that, given a suitable notation, a unifying analytic
algorithm was almost bound to follow. Even Huygens, who in his
earlier work had scrupulously followed the methods of the ancients,
was after 1655 au courant with the new points of view and made fre-
quent use of them. He attempted a generalization of the methods of
the ancients, in the manner of Valerio; 310 he repeated the infinitesimal
demonstrations of Galileo and Torricelli on falling bodies; 311 he showed
the influence of Cavalieri in speaking of lines as elements of surfaces ; 312
and his work includes frequent applications of Fermat's methods of
tangents and maxima and minima. 313 Huygens, however, was a mathe-
matical classicist, and it remained for his two younger friends, Newton
and Leibniz, to bind all of this work into what represents probably the
most effective instrument for scientific investigation that mathematics
has ever produced. 314
804 Sluse, "A Method of Drawing Tangents to All Geometrical Curves," p. 38.
m Huygens, (Euvres completes, XIV, 446-47.
*» Ibid., XIV, 442-48, 504-17.
** See the second letter by Hudde in Geometria, a Renoto des Cartes anno 1637 Gattice
edita, I, 507-16.
810 (Euvres completes, XIV, 338-39. m Ibid., XVI, 114-18.
m Ibid., XI, 158; XII, 5; XIII, 753; XIV, 192, 337.
m Ibid., XI, 19; XVI, 153; XIV, passim.
314 Cf. Bell, The Queen of the Sciences, p. 8.
V. Newton and Leibniz
FEW NEW branches of mathematics are the work of single
individuals. The analytic geometry of Descartes and Fermat was
certainly not the result of their investigations only, but was the
outgrowth of several mathematical trends which converged in the
sixteenth and seventeenth centuries. It was the result of the influence
of Apollonius, Oresme, Viete, and many others.
Far less is the development of the calculus to be ascribed to one or
two men. We have followed the long and uneven flow of thought which
led from the philosophical speculations and the mathematical demon-
strations of the ancients to the remarkably successful heuristic
methods of the seventeenth century. We have indicated that the
procedures invented by Fermat, for example, are almost identical
with those found in the calculus, and that the new propositions dis-
covered by Barrow include the geometrical equivalent of the basic
theorem of the subject.
The time was indeed ripe, in the second half of the seventeenth
century, for someone to organize the views, methods, and discoveries
involved in the infinitesimal analysis into a new subject characterized
by a distinctive method of procedure. Fermat had not done this,
largely because of his failure to generalize his methods and to recog-
nize that the problems of tangents and quadratures were two aspects
of a single mathematical analysis — that the one was the inverse of
the other. Barrow was unable to establish the new subject for, al-
though the first to recognize clearly the unifying significance of this
inverse property, 1 he failed to realize that his theorems were the basis
for a new subject. Being unsympathetic with the Cartesian mathe-
matical analysis and the algebraic trend, he implied that his results
were to be considered as rounding out the geometry of the ancients. 2
The traditional view, therefore, ascribes the invention of the cal-
culus to the more famous mathematicians, Isaac Newton and Gott-
fried Wilhelm von Leibniz. From the point of view of the development
of the concepts involved, the aspect which concerns us chiefly here,
1 Geometrical Lectures, p. 124. * Ibid., p. 66.
188 Newton and Leibniz
it might be far better to speak of the evolution of the calculus. Never-
theless, inasmuch as Newton and Leibniz, apparently independently,
invented algorithmic procedures which were universally applicable
and which were essentially the same as those employed at the present
time in the calculus, and since such methods were necessary for the
later logical development of the conceptions of the derivative and the
integral, there will be no inconsistency involved in thinking of these
men as the inventors of the subject. In doing so, however, we are not
to consider or to imply that they are responsible for the ideas and defi-
nitions underlying the subject at the present time; for these basic
notions were to be rigorously elaborated only after two centuries of
further effort in this direction. Furthermore, inasmuch as we are here
more concerned with ideas than with rules of procedure, we shall not
discuss the shamefully bitter controversy 3 as to the priority and
independence of the inventions by Newton and Leibniz. 4 Both men
owed a very great deal to their immediate predecessors in the develop-
ment of the new analysis, and the resulting formulations of Newton
and Leibniz were most probably the results of a common anterior,
rather than a reciprocal coincident, influence.
Attempts have been made by historians of the calculus 5 to trace
two distinctly different threads of development: one, the kinematic,
leading to Newton through Plato, Archimedes, Galileo, Cavalieri, and
Barrow; and the other, the atomistic, tending toward Leibniz through
Democritus, Kepler, Fermat, Pascal, and Huygens. There is, how-
ever, a complete lack of recognition of such a cleavage by the mathe-
maticians involved, nor can we now distinguish the views and methods
of the one "group," throughout the seventeenth century, from those
1 See Hathaway, "The Discovery of Calculus," pp. 41-43; and "Further History of the
Calculus," pp. 166-67, 464-65, for accusations against Leibniz of "the foundation of a plot
to deprive Newton of all credit . . . , with typical German propaganda," and of inaugu-
rating "that system of espionage on scientific work in foreign countries by which the
usefulness and credit of as much of that work as possible might be transferred to Germany."
4 See Sloman, The Claim of Leibnitz to the Invention of the Differential Calculus; Leibniz,
The Early Mathematical Manuscripts (ed. by Child) for the statement of the suspicions
directed against Leibniz. For his defense see Gerhardt's two works, Die Entdeckung der
Differ entialrechnung durch Leibniz, and Die Entdeckung der hoheren Analysis; and also
Mahnke's two articles, "Neue Einblicke in die Entdeckungsgeschichte der hoheren
Analysis," and "Zur Keimesgeschichte der Leibnizschen Differentialrechnung." For an
extensive chronological bibliography on the subject of the controversy, see De Morgan,
Essays on the Life and Work of Newton.
6 See, e. g., Hoppe, "Zur Geschichte der Infinitesimalrechnung," pp. 175-76.
Newton and Leibniz 189
of the other. Galileo, Cavalieri, Torricelli, and Barrow used both
fluxionary and infinitesimal considerations, and the procedures of
Fermat, Pascal, and Huygens were perhaps as well known to Newton
and the English mathematicians as to Leibniz. The geometric devel-
opments leading to the fluxionary calculus of Newton were not es-
sentially other than those pointing the way toward the differential
calculus of Leibniz. However, after the methods of procedure of the
subject had been established, and the logical and metaphysical bases
of these were brought into question, the contrast between the points
of view and the modes of presentation became heightened by the
contrasting scientific and philosophic tastes of the inventors, as well,
perhaps, as by the priority controversy in which blind loyalties
prevented their successors from appreciating the advantages and dis-
advantages of the two systems. We shall attempt, therefore, to point
out here not only the origins of the work of Newton and Leibniz, but
also the nature of the interpretations which they later gave and the
significance of these for the development of the fundamental notions
of the calculus.
Isaac Newton was the student, at Cambridge, of Isaac Barrow and
so came strongly under the influence of the latter, whose Geometrical
Lectures he helped to prepare for publication. Now JBarrow jyas
familiar with the work of Cavalieri and the two views of the genera-
tion of geometrical magnitudes there presented — that of indivisibles,
and that of flowing quantities. He thought of a tangent to a curve not
only as the prolongation of one of the infinitely many lineal elements
of which the curve might be assumed to be composed, but also as the
direction of motion of a point which, by moving, generated the curve.
These views were almost certainly familiar to Newton through his
attendance at Barrow's lectures. Barrow, however, lacked an apprecia-
tion of the analytical methods of Descartes and Fermat, and failed
to realize the significance of Wallis' arithmetization. These newer
views, however, had been presented in 1655 by the works of Wallis,
to which we have already referred and with which Newton became
acquainted during the period of his early mathematical training. 6
Newton, in fact, acknowledged that he had been led to his first dis-
coveries in analysis and fluxions by the Arithmetica infinitorum of
6 Sloman, The Claim of Leibnitz to the Invention of the Calculus, pp. 1-7.
190 Newton and Leibniz
Wallis, 7 and the principles of induction and interpolation which
Wallis there employed may have been instrumental also in leading
Newton to the discovery of the binomial theorem. 8 Newton's concep-
tion of number resembles that of Wallis rather than"that of Barrow-
less a collection of units than an abstract ratio of any quantity to
another, a definition which also includes irrational ratios as numbers. 9
Newton in this respect went beyond Wallis and Descartes in regarding
negative ratios as numbers in the true sense of the word 10 — a gen-
eralization of the geometrical representation of Descartes.
Another element adding to the effectiveness of Newton's presenta-
tion of the method of fluxions was the use of infinite series. The
Scholastic philosophers had studied infinite series in connection with
the geometric representation of variability, and Gregory of St. Vin-
cent, Tacquet, and Fermat had made use of infinite progressions.
However, the earliest investigations of general arithmetic infinite
series were largely the work of English mathematicians such as
Wallis and James Gregory. Incidentally, the important work in this
connection by the latter appeared only a year or two before Newton
composed his first treatise on the calculus — one in which he employed
infinite series in connection with the binomial theorem. The use of
such series did indeed make for a universality of application of the
method of fluxions and aided in freeing it from geometrical prejudices,
but there has been a tendency on the part of historians to focus
attention upon Newton's use of infinite series, rather than upon other
more essential aspects of his work. 11
/ Newton tells us that he was in possession of his fluxionary calculus
\ as early as 1665-66, 12 that is, at some time during the period in which
he had heard Barrow's lectures and had discovered the binomial the-
orem. The first notice of his calculus was given, however, in 1669, in
De analyst per aequationes numero terminorum infinitas. 13 This was
not published until 1711, but it circulated among his friends. In this
7 More, Isaac Newton, p. 184.
8 Merton, "Science, Technology and Society in Seventeenth Century England," p.
472, n.
8 Newton, Opera omnia, I, 2.
10 Ibid., I, 3. Cf. Schubert, "Principes fondamentaux de l'arithmetique," pp. 35-37.
11 Cf. Zeuthen, "Notes," 1895, pp. 194 ff., and a review of these by Tannery in Bulletin
de Darboux, 2d series, XX (1896), 24^28.
12 Opera omnia, I, 333. M Opera omnia, I, 257-82; Opuscula, I, 3-28.
Newton and Leibniz 191
monograph he did not explicitly make use of the fluxionary notation
or idea. Instead, he used the infinitely small, both geometrically and
analytically, in a manner similar to that found in Barrow and Fermat,
and extended its applicability by the use of the binomial theorem. In
this paper Newton employed the idea of an indefinitely small rectangle
or "moment" of area and found the quadratures of curves as follows:
Let the curve be so drawn that for the abscissa x and the ordinate y
(\ m +n
1 ax n . Let the moment or infinitesimal
m + nj
increase in the abscissa, following the notation of James Gregory,
be o. The new abscissa will then be x + o and the augmented area
I n \ ' !L±Jt
z + oy = ( I a(x + o) n . If in this expression we apply the
\m + n)
binomial theorem, divide throughout by o, and then neglect the
m
terms still containing o, the result will be y = ax n . That is, if the
^ m + n *»
ft -
area is given by z = ax n , the curve will be y = ax n .
m -f- n
- n ^^
Conversely, if the curve is y = ax n , the area will be z = ax * , li
m-\-n
Here we have an expression for area which was arrived at, not
through the determination of the sum of infinitesimal areas, nor
through equivalent methods which had been employed by Newton's
predecessors from Antiphon to Pascal. Instead, it was obtained by a
consideration of the momentary increase in the area at the point in
question. In other words, whereas previous quadratures had been
found by means of the equivalent of the definite integral defined as
a limit of a sum, Newton here determined first the rate of change of
the area, and then from this found the area itself by what we should
now call the indefinite integral of the function representing the
ordinate. It is to be noted, furthermore, that the process which is
made fundamental in this proposition is the determination of rates
of change. In other words, what we should now call the derivative
is taken as the basic idea and the integral is defined in terms of this.
Mathematicians from the time of Torricelli to Barrow had in a sense
known of such a relationship, but Newton was the first man to give
14 Opera omnia, I, 281 ; Opuscula, I, 26.
192 Newton and Leibniz
a generally applicable procedure for determining an instantaneous rate
of change and to invert this in the case of problems involving sum-
mations. Before this time the tendency had been rather in the opposite
direction — to reduce problems, whenever possible, to the determina-
tion of quadratures. With this step made by Newton, we may consider
that the calculus has been introduced.
Newton applied this method to the quadrature of numerous curves,
f a 2
such as y — x 2 -f- x and y = . A few years later, in sending
b -f- x
these results to Collins, he described in addition a number of proposi-
tions in maxima and minima and in tangents, which he had obtained
by means of his methods. It is this letter to Collins of December 10,
1672, which became significant in the controversy as to whether
Leibniz made his discoveries independently of Newton. In this letter
Newton pointed out frankly that his rules are analogous to those of
Sluze and Hudde, although more general; 16 and in another place he
admitted that he had gotten the hint for his procedure from the
method of Fermat, improved by Gregory and Barrow. 16 The change
from the E of Fermat to the o of Gregory and Newton is, of course,
trivial. It has from time to time been interpreted as a substitution of
zero for E, a view which would reduce Newton's method to a mean-
ingless manipulation of zeroes, somewhat in the manner of Bhaskara. 17
Newton distinctly regarded his symbol as the letter o and not the
cipher zero, and in this respect it is entirely comparable to Fermat's
E. The significance of Newton's work lay first of all in the fact that
he applied the method "directly and invertedly," as he said. 18 In the
second place Newton regarded it, in connection with the use of infinite
series, as a universal algorithm, whereas that of Fermat, as well as the
modifications of this by Sluse, Hudde, and Huygens, availed only in
the case of rational algebraic functions.
It will be noticed that although the work of Newton contains the
essential procedures of the calculus, the justification of these is not
16 Opera omnia, IV, 510; cf. also Mathematical Principles (Cajori), pp. 251-52.
14 More, Isaac Newton, p. 185, n.
17 The entire interpretation of Newton given by Hoppe, "Zur Geschichte der Infinitesi-
malrechnung," is vitiated by the fact that he has followed certain older historians in this
mistake. Cf. Gerhardt, Die Entdeckung der hbheren Analysis, p. 80; Weissenborn, Die
Principien der hbheren Analysis, p. 25, n.; Gerhardt, "Zur Geschichte des Streites," p. 131.
u More, loc. cit.
Newton and Leibniz 193
clear from the explanation he gave. Newton did not point out by what
right the terms involving powers of o were to be dropped out of the
calculation, any more than Fermat justified omitting the powers of
E, or Barrow those of e and a. His contribution was that of facilitating
the operations, rather than of clarifying the conceptions. As Newton
himself admitted in this work, his method is "shortly explained
rather than accurately demonstrated."
m
In his demonstration above that the area of y = ax* is given by
n "-^
z = ax n , however, we can see some hint of the thought in
n + m
his mind. The ordinate y seems to represent the velocity of the increas-
ing area, and the abscissa represents the time. Now the product of the
ordinate by a small interval of the base will give a small portion of the
area, and the total area under the curve is only the sum of all of these
moments of area. This is exactly the infinitesimal conception of Oresme,
Galileo, Descartes, and others, in their demonstrations of the law of
falling bodies, except that these men had found the area as a whole
through the addition of such elements, whereas Newton found the
area from its rate of change at a single point. It is difficult to tell in
exactly what manner Newton thought of this instantaneous rate of
change, but he very likely accepted it as similar to the conception of
velocity which Galileo had made so familiar but had not defined
rigorously. A thorough-going empiricist for whom mathematics was a
method rather than an explanation, 19 Newton apparently considered
any attempt to question the instantaneity of motion as linked with
metaphysics, and so avoided framing a definition of it. Nevertheless,
he accepted this notion and made it the basis of his second and more
extensive exposition of the calculus, as given in the Methodus fluxionutn
et serierum infinitarum, 20 which was written about 167 1, 21 but not
published until 1736.
In this book Newton introduced his characteristic notation and
conceptions. Here he regarded his variable quantities as generated by
the continuous motion of points, lines, and planes, rather than as
aggregates of infinitesimal elements, the view which had appeared in
19 Burtt, Metaphysical Foundations, pp. 208-10.
20 Opusctda, I, 31-200.
21 See Zeuthen, "Notes," 1895, p. 203; cf. also Newton, Opera omnia, II, 280.
194 Newton and Leibniz
De analyst. Just as Barrow found the chief characteristic of time in
its even flow, so also his pupil Newton, although he did not "formally
consider time," 22 was influenced to make continuous motion funda-
mental in his system. This concept Newton seems to have felt was
sufficiently impelling and so clearly known through intuition as to
make further definition unnecessary. The rate of generation Newton
called a fluxion, designating it by means of a letter with a dot over it,
a "pricked letter"; the quantity generated he called a. fluent, employ-
ing in this connection the terms which had appeared earlier in the
work of Calculator. Thus if x and y are the fluents, then their fluxions
are x and y. Incidentally, Newton in other places 23 proceeded to point
out that one may consider the fluxions x and y, in turn, as fluents of
which the fluxions are represented by 'x and y, and so on. The fluents
of which x and y are the fluxions Newton represented by x and y;
the fluents of which these latter quantities are the fluxions were
written x and y, and so on.
In the Methodus fluxionum Newton stated clearly the fundamental
problem of the calculus: the relation of quantities being given, to
find the relation of the fluxions of these; and conversely. 24 In con-
formity with this problem and the new notation, Newton then gave
examples of his method. These may be represented by the determina-
tion of the fluxion of y = x 11 . His approach in this case is but slightly
different from the earlier exposition in De analyst. If o is an infinitely
small interval of time, then xo and yo will be the indefinitely small
increments, or moments, of the flowing quantities x and y. In y = x n
one then substitutes x + xo for x and y -f- yo for y, expands as before
by the binomial theorem, cancels the terms not containing o, and
divides throughout by o. Since, moreover, o was assumed to be
infinitely small, the terms containing this — that is, the moments of
quantities — can be considered as zero in comparison with the others,
and are to be neglected. 28
The result, y = nx n ~ i x, is, of course, the same as that obtained
by Newton previously in the De analyst without the use of fluxions.
22 Opuscula, I, 54.
23 Cf. Opera omnia, I, 338; Opuscula, I, 208. M Opuscula, I, 55, 61.
26 Opuscula, I, 60. The illustration, y = x", which we have presented is not here given
in this form by Newton, but is taken as representative, in form and argument, of the
exercises he gave.
Newton and Leibniz 195
It is to be noted that the introduction here of the conception of a fluxion
is not an essential modification of the earlier work. The infinitely small
enters as persistently as in the 1669 exposition, but in the dynamic
form of Galileo's moment or the conatus of Hobbes rather than in
the static form of Cavalieri's indivisible. This change serves only in
intuition to remove the harshness (as Newton expressed it) from the
doctrine of indivisibles. 26 In thought the justification of the neglect
of infinitely small terms is to be made on precisely the same basis,
whether it be written E, e, a, o, or ox. Newton himself seemed to feel
here some need for the limit concept, for he pointed out that fluxions
are never considered alone, but always in ratios. 27 Later, when Newton
sought escape from the clutches of the infinitely small, he emphasized
this fact much more strongly.
This third stage in his thought appears clearly in De quadratura
curvarum, 28 which was written in 1676 but not published until 1704.
In this treatise Newton sought to remove all traces of the infinitely
small. Mathematical quantities were not to be considered as made
up of moments or very small parts, but as described by continuous
motion. In determining the fluxion of x n , Newton proceeded much as
in the Methodus fluxionum, replacing x by (x + o). In conformity
with the fluxionary symbolism it would be expected that the incre-
ment in x should be designated ox instead of o, but inasmuch as
Newton is here dealing with only a single variable, the fluxion of this
may conveniently be taken as unity. On expanding (x + o) n by the
binomial theorem, and subtracting x n , the result is, of course, the
change in x n corresponding to the change o in x. Instead, now, of
completing the argument by a doubtfully justified neglect of terms,
Newton formed the ratio of the change in x to the change in x n :
that is, 1 to nx n ~ 1 + n I J ox n ~ 2 + . . . ; and in this he allowed
o to approach zero — to vanish. The resulting ratio, 1 to nx n ~ *, we
should speak of as the limit of the ratio of the changes, but Newton
called it the ultimate ratio of the changes — a terminology which
was later to lead to some confusion in thought. This ultimate ratio
of "evanescent increments" is the same as the prime or first ratio of
28 Opera omnia, I, 250; II, 39.
27 Opuscula, I, 63-64. M Opera omnia, I, 333-86; Opusculo, I, 203-44.
196 Newton and Leibniz
the "nascent augments." It is likewise the ratio of the fluxions at the
point in question. 29
In the above demonstration, the essential elements of the derivative
are more clearly present than in any other part of Newton's work:
the emphasis upon a function of one variable, rather than upon an
equation in several; the formation of the ratio of the changes in the
independent variable and in the function; and, finally, the deter-
mination of the limit of this ratio as the changes approach zero.
Incidentally, the ratio as expressed by Newton is commonly inverted
in the modern derivative. There are in Newton's thought, moreover,
certain elements which have since been discarded as adscititious: his
appeal to time as an auxiliary independent variable is now considered
gratuitous; and the limiting ratio is now regarded as a single number,
rather than as the quotient of two rates of change. Had Newton
devoted more of his time to clarifying the elements of thought in his
demonstration by ultimate ratios, the calculus might have been
established upon the concept of the derivative a century before the
time of Cauchy. In his first published account of his new analysis
Newton suggested this type of argument; but in his illustrations of
the method of fluxions in this work he unfortunately resorted to the
infinitesimal terminology of his earlier accounts.
Newton's discovery of the calculus dates back to the years 1665
and 1666, as he says in De quadrature/,. Within the following decade
he had written out, as we have seen, three accounts of his methods,
but had published nothing. By 1676 he became aware that Leibniz
was working on similar problems, and on October 24 of this year
he sent a letter to Leibniz, through Oldenburg, in which he gave in
the form of an anagram a statement of the fundamental problem
of his calculus. This seems to have been his only effort to assure his
claim to priority in the invention of the calculus. Upon transposing
the letters of this anagram and translating, it read: "Given in an
equation the fluents of any number of quantities, to find the fluxions
and vice versa." 30 Similar statements of the problem of the calculus
had been included in the Methodus fluxionum and the De quadratura
which he had already composed. 31
" Opera omnia, 1, 334.
n Opera omnia, IV, 540 ff.; cf. also Leibniz, Mathematische Sckrifkn, I, 122-47.
u Opera omnia, I, 339, 342; Opuscule, I, 55, 61.
Newton and Leibniz 197
In this letter he also admitted his indebtedness to Wallis, James
Gregory, Sluse, and others, but did not give an exposition of his
methods. The first published account of his calculus appeared some-
what incidentally, more than ten years later, in the famous Principia
mathematica philosophiae naturalis of 1687. The propositions in this
book, concerned as they are with velocities, accelerations, tangents, and
curvatures, are largely those handled now by the methods of the
calculus, but Newton presented them in the form of synthetic geo-
metrical demonstrations with an almost complete lack of analytical
calculations. Nevertheless, Newton at several points in the work gave
indications of more general points of view.
In a series of lemmas in the first book, he expressed the type of
argument appearing in De quadratura curvarum. "Quantities, and the
ratios of quantities, which in any finite time converge continually to
equality, and before the end of that time approach nearer to each
other than by any given difference, become ultimately equal." 32 This,
of course, is the sort of general limit proposition which Stevin, Valerio,
Gregory of St. Vincent, Tacquet, Wallis, and others had attempted to
substitute for the Greek method of exhaustion. In fact a passage in
Gregory of St. Vincent, 33 in which the word "terminus" was used to
designate the limit of a progression, may have been the origin of the
term "ultimate ratio" which Newton was to use so frequently. New-
ton's view of a limit, like that of these earlier workers, was bound
up with geometric intuitions which led him to make vague and
ambiguous statements. Thus he said, "The ultimate ratio of the arc,
chord, and tangent, any one to any other, is the ratio of equality." 34
and a little later he spoke of the similarity of the "ultimate forms of
evanescent triangles." 36
These remarks imply that Newton was not thinking arithmetically,
as we do now, of the limit of the sequence of numbers representing
the ratios of the (arithmetical) lengths of the geometrical quantities
involved, as these become indefinitely small, but that he also was
influenced by the infinitesimal views of the seventeenth century to
think of ultimate geometrical indivisibles. It is true that he never
used the expressions ultimate arcs, chord, tangents, or triangles, but
only those of ultimate ratios and forms, expressions which allow of
M Opera omnia, I, 237; II, 30. M Opus geometricum, p. 55.
« Opera omnia, I, 242; II, 34. » Ibid., I, 243.
198 Newton and Leibniz
rigorously correct abstract interpretations, but which strongly sug-
gest others in terms of the intuitively more attractive view afforded
by infinitesimals. That Newton realized the difficulties involved in a
naive view of infinitesimals is indicated, however, in his further
statement in the Principia that "Ultimate ratios in which quantities
vanish, are not, strictly speaking, ratios of ultimate quantities, but
limits to which the ratios of these quantities decreasing without limit,
approach, and which, though they can come nearer than any given
difference whatever, they can neither pass over nor attain before the
quantities have diminished indefinitely." 36 This is the clearest state-
ment Newton gave as to the nature of ultimate ratios, but we shall
find that, in continuing this argument in a lemma in the second book
of his Principia, his exposition again took on more strongly the
dependence upon the idea of infinitely small quantities, with the con-
cept of limit somewhat hazily implied as basic. It is precisely this
lack of arithmetical clarity which led, in the following century, to
controversial discussions, not only on the validity of Newton's flux-
ions, but also as to what Newton really meant by the above statements
and others similar to them.
Inasmuch as the Principia is written in the old synthetic geometric
manner, references to the method of fluxions are not numerous. In
the second book, however, there appeared the first publication of
"the foundation of that general method." 37 Here one finds the state-
ment of the fundamental principle, "The moment of any genitum is
equal to the moments of each of the generating sides multiplied by
the indices of the powers of those sides, and by their coefficients con-
tinually." Newton proved this first for the product AB as follows:
Let AB represent a rectangle and let the sides A and B be diminished
by \a and \b respectively. The diminished area will then be AB —
\aB — \bA -f- \ab. Now let the sides of AB be increased by \a and
\b respectively. The area of the enlarged rectangle will then be AB
-f- \aB -f \bA + \ab. Subtracting the smallest rectangle from the
largest, one obtains aB -f bA as the moment of the original rectangle,
corresponding to the moments a and b of A and B; which proves the
proposition for this product. If A = B, the moment of A 2 is deter-
mined in turn as 2aA .
*• Opera omnia, I, 251. » Opera omnia, II, 277-80.
Newton and Leibniz 199
By the use of the decrements \a and \b and the increments \a
and \b, instead of the increments a and b, Newton here avoided the
necessity of dropping the infinitely small term ab. Newton thus made
explicit use of infinitely small quantities of first order only — in which
respect his work is to be contrasted with that of Leibniz — but his
procedure in the proposition above was later justly criticized as
implying the omission of infinitesimals of second order.
To find the moment of ABC Newton let AB = G and, by applying
the first part of the theorem, obtained cAB + bCA + aBC as the
result. Letting A = B = Cj the moment of A 3 is in turn determined
as 3aA 2 . By similar procedures the moment of A" for positive integral
powers is found to be naA n ~ l . This same result is seen to hold for
negative powers also. This is apparent from the following considera-
tions. Let m be the moment of — . Then from — - A = \ and the
A A
1 a
moment of a product, one obtains -•c + i.w = 0,orm= — .
A A 2
This argument is easily generalized to include all negative integral
powers, and, with slight modifications, it is applicable to all*products
of rational powers of variables.
Newton said 38 that this is the foundation of his method of tangents
and quadratures; and, in fact, this rule, combined with the use of
infinite series, is sufficient for arriving at the essential results of the
method of fluxions. However, because Newton supplied here such an
unfortunately brief exposition of his procedure and its justification,
and inserted this short account; in . the second book in the un-
obtrusive form of a lemma 39 to oflier propositions, some doubt
has been expressed as to the seriousness with which it was put
forward. 40
The basis of the calculus as thus first published in the Principia,
is, of course, to be found in the nature of Newton's moments; but
it is just here that Newton was very far from clear in his language.
He said on this point, "Finite particles are not moments, but the
88 Ibid. » Opera omnia, II, 277-80.
"Moritz Cantor (Vorlesungen, III, 192) would put but little emphasis upon the
lemmas in the Principia, whereas Zeuthen ("Notes," 1895, p. 249; cf. also Geschichte der
Mathematik im XVI. and XVII. Jahrhundert, pp. 382-84) stresses their significance in the
calculus.
200 Newton and Leibniz
very quantities generated by the moments. We are to conceive them
as the just nascent principles of finite magnitudes." Perhaps realizing
that this statement made his moments as vague as the infinitesimals
of Cavalieri, Fermat, and Barrow, he justified himself by adding,
"Nor do we in this Lemma regard the magnitude of the moments,
but their first proportion as nascent." 41 This looks like an attempt
to bring in the doctrine of limits, which he had formulated in Book I,
in which he regarded the ratio as ultimate, without specifying that
the quantities entering it were so. Nevertheless, it is difficult to see
just how one is to think of the limit of a ratio in determining the
moment of AB. We have to deal here with two variables and are faced
with the equivalent of partial differentiation, unless we have recourse
to time as a single independent variable, as Newton next suggested.
Perhaps realizing the difficulties in the way of interpreting the proposi-
tion in terms of the ratio or proportion of infinitesimals, Newton added
another interpretation. "It will be the same thing, if, instead of
moments, we use either the velocities of the increments and decre-
ments (which may also be called the motions, mutations, and fluxions
of quantities), or any finite quantities proportional to those veloc-
ities." 42
To summarize the above, we see that Newton first had in mind
infinitely small quantities which are not finite nor yet precisely zero.
"Ghosts of departed quantities" they were fittingly called by the
critics of the method in the following century. These offer too great
difficulty of conception, so Newton next focused attention on their
ratio, which in general is a finite number. Knowing this ratio, one
may now substitute for the infinitesimal quantities forming it any
other easily conceived finite magnitudes having the same ratio, such
as quantities which are thought of as the velocities or fluxions of
those entering into the equation. Newton thus offered in the Principia
three modes of interpretation of the new analysis: that in terms of
infinitesimals (used in his De analysi, his first work) ; that in terms of
prime and ultimate ratios or limits (given particularly in De quad-
ratures, and the view which he seems to have considered most rigor-
ous) ; and that in terms of fluxions (given in his Methodus fluxionum,
and the one which appears to have appealed most strongly to his
41 Opera omnia, II, 278. <* Ibid.
Newton and Leibniz 201
imagination). The fact that Newton could thus present all three views
as essentially equivalent shows us how far he was from viewing his
method as quite distinct from the somewhat equivalent methods of
his predecessors and contemporaries. In De guadratura, after saying
that he there considered quantities as described by a continued
motion and used the method of prime and ultimate ratios, he asserted
that his method is consonant to the geometry of the ancients; 43 and
in the Principia he also admitted that Leibniz had a similar method
for considering the generation of magnitudes — an admission which
was, however, omitted from later editions. 44 In fact the method of
fluxions is dependent upon some other method, such as limits or
infinitesimals, for the determination of the basic relations between
the fluxions. Although Newton apparently preferred to link his
method of fluxions with the idea of a limiting ratio, he so often used
infinitesimals for dispatch that we shall find many of his successors
later interpreting the fluxions themselves as infinitely small quan-
tities, confusing them with moments.
Newton himself frequently used the concept of the infinitely small
throughout his early work, but tended to become wary of it in his later
expositions. In a portion of De guadratura which appeared in Wallis's
Algebra of 1693, Newton had said that terms multiplied by o he omitted
as infinitely small, thus obtaining the result. 46 In the 1704 publication
of the work, on the other hand, he said clearly, that "errors are not
to be disregarded in mathematics, no matter how small." 46 The con-
clusion was to be reached, not by simply neglecting infinitely small
terms, but by finding the ultimate ratio as these terms become evan-
escent. However, even after this he did not abjure the infinitesimal
completely, but continued to speak of moments as infinitely little
parts. Furthermore, Newton added to the confusion in the thought
of his contemporaries on fluxions by failing sometimes to multiply the
fluxions by o when he wished to represent moments. Although he said
that wherever pricked letters represent moments and are without the
letter o, this letter is always to be understood, very many English
mathematicians began to associate fluxions with the infinitely small
« Opera omnia, I, 338. ** Ibid., H, 280.
« See De Morgan, "On the Early History of Infinitesimals in England," p. 324. See
also Cajori, "Newton's Fluxions," p. 192, and Raphson, The History of Fluxions, p. 14.
46 Opera omnia, I, 338.
202 Newton and Leibniz
differentials of Leibniz. 47 Newton's final view of the basis of the
subject, however, seems to be that shown in his remark in De quad-
rature "I have sought to demonstrate that in the method of fluxions
it is not necessary to introduce into geometry infinitely small
figures. 48
We have seen that most of Newton's work on the calculus was
written in the period from 1665 to 1676, but none of it had been pub-
lished during that time. It has been suggested 49 that Newton's long
delay in the publication of his three chief works on the calculus was
occasioned by the fact that he was dissatisfied with the logical founda-
tions of the subject. In the meantime, however, other mathematicians
were looking for the general principle needed to solve the problems
of tangents, maxima and minima, and quadratures. The methods of
Fermat had already been modified by Huygens, Hudde, Sluze, and
others. These men were contemporaries of the most versatile genius
of the period, Gottfried Wilhelm von Leibniz, who, like Newton, was
to develop rules and a literal symbolism for putting all of the infinites-
imal considerations together under an algorithmic procedure. Although
^interested primarily in law and logic, Leibniz had in his early days
written a little on arithmetic and mechanics. In 1672, however, he
met Huygens in Paris and was urged by him to make a deeper study
/of mathematics. On a visit to London, in 1673, he met a number qL\-
mathematicians^ learned much about infinite series, purchased a copy j
of Barrow's Lectures, and may have/known, through Collins, of j
\ Newton's De analyst. After his return to Paris in the same year, ^e '
studied the mathematical works ol Cavalieri, Torricelli, Gregory
of St. Vincent, Roberval, Pascal, Descartes, Wren, James Gregory,
Sluse, Hudde, and others. 60 The background of Leibniz in infinitesimal
analysis was thus not greatly different from that of Newton; for the
results of these men were well known in England as well as on the
Continent. The early mathematical reading of Leibniz was thus largely
on geometry, but he hapVother interests also which may have been
y
^Montucla (Histoire des mathematiques, II, 373) misinterprets JJewton's occasional
omission of these letters as an indication of confusion in Newton's thought between
velocity and increment.
18 Opera omnia, I, 333; Opuscula, I, 203.
49 Merz, A History of European Thought in the Nineteenth Century, II, 630.
60 See Gerhardt, Die Entdeckung der Differentialrechnung durch Leibniz, p. 31 and
passim.
Newton and Leibniz 203
decisive in shaping his analysis. His first mathematical paper had been
on combinatorial analysis, and he always retained a strong arith-
metical tendency. One of the first fruits of his study of problems in
quadratures was the "Arithmetical Tetragonism," in which he found
the area of a unit circle to be given by four times the infinite series
1 _ i + £ — j. -j- ... .»i These formalistic and arithmetical con-
siderations were now to combine in an interesting way with the
geometry which Leibniz had begun to master.
During his study at this time Leibniz was working on the problem
of tangents, as well as that of quadratures, and had reached a solu-
tion based upon the "characteristic triangle" — the differential triangle
which had appeared in various forms, particularly in the works of
Torricelli, Fermat, and Barrow. It is difficult to determine the filiation
of events leading to the invention by Leibniz of the differential cal-
culus, but he himself, in a letter written thirty years later, attributed
the inspiration for his use of the differential triangle to a figure (fig. 18
above) he had run across, about 1673, in Pascal's Traite des sinus du
quart de cercle. 62 Leibniz said that on the reading of this example in
Pascal a light suddenly burst upon him and that he then realized
what Pascal had not — that the determination of the tangent to a
curve depended upon the ratio of the differences in the ordinates and
abscissas, as these became infinitely small, and that the quadrature
depended upon the sum of the ordinates, or infinitely thin rectangles,
for infinitesimal intervals in the abscissas. Moreover, the operations
of summing and of finding differences were mutually inverse. Barrow
had, in a sense, realized this also, for his a and e method for tangents
involved the differences of ordinates and abscissas, his quadratures
were effected by the summation of infinitesimals, and his inversion
theorem showed the relationship of the two problems; but he had
never developed these into a unified procedure. Leibniz, on the other
hand, continued studying the characteristic triangle, encouraged by
61 See Mathematische Sckriften, V, 88; cf. also Leibniz, Early Mathematical Manuscripts,
p. 163.
M Gerhardt and Zeuthen stress the obligation of Leibniz to Pascal. Child feels that
inasmuch as Leibniz would have found in Barrow anticipations of the differential triangle
much clearer than that in Pascal, he may have failed to mention Barrow either through a
desire not to point out his indebtedness or through having forgotten the influence Barrow
had had upon him. See especially Leibniz, Early Mathematical Manuscripts, pp. 15-16,
for a discussion of the diagrams involved on this point; cf. also the works of Gerhardt and
Zeuthen cited in the bibliography.
204
Newton and Leibniz
Huygens, and related this work to his former interest in combinatorial
analysis. 53
In the harmonic triangle and in the arithmetic triangle of Pascal,
there are striking relationships. For example, in the arithmetic tri-
Arithmetic Triangle Harmonic Triangle
111111... * 4 4 * 4 1 . . .
12 3 4 5... *
1 3 6 10 . . .
1 4 10 . . .
1 5 . . .
1 . . .
i
i
1
■sir
TV
i
*V
TIT
ifV
7TT •
angle any element is the sum of all of the terms in the line above it
and to the left, and it is also the difference between the two terms
directly below it. Similarly, in the harmonic triangle any element is
the sum of the terms in the line below and to the right, and it is also
the difference between the two terms just above it.
That is, in Pascal's arithmetic triangle, if we designate by x the
numbers in any one line, the numbers of the first line below will be
the sum of all the x's up to this point, reading from left to right;
those in the second line below will be the sum of the sums of all the
x's, and so on. Conversely, the lines above represent the differences
the differences of the differences, and so on. Similarly, in the har-
monic triangle, with respect to the elements of any line, those below
represent the differences, the differences of the differences, and so on;
those above are the sums, the sums of the sums, and so on — reading
from the right, however. Thus in these triangles we see that sums and
differences are the inverses of each other, 64 just as the problem of
tangents, which depends on the differences of ordinates, is the inverse
of that of quadratures, which depends on the sum of all the ordinates,
in the sense of Cavalieri. However, whereas the differences between
elements in the arithmetic and harmonic triangles are finite, those
between the ordinates of a curve are infinitesimal, and the formulas
applicable in the former case no longer hold in the case of curves.
63 See Mathematische Schriften, V, 108, 404-5; see also Newton, Opera omnia, IV, 512-15,
and Gerhardt, Die Entdeckung der Differentialrechnung dutch Leibniz, pp. 54-56.
M See Leibniz, Early Mathematical Manuscripts, p. 142, and Mathematische Schriften,
V, 397.
Newton and Leibniz 205
It was therefore necessary for Leibniz to develop a method of
procedure for determining sums and differences of infinitesimals. This
he appears to have done by about 1676, the time at which Newton
composed De quadrate ra. He had, about a year before, adopted his
characteristic notation. He employed $x, or later fxdx, for the "sum"
of all the x's — or the "integral" of x, as he called it later, on the
suggestion of the Bernoulli brothers. 55 For the "differences" in the
x
values of x, he wrote dx, although he had at first used - for this, in
d
order to imply that finding the "difference" involved a lowering of
the dimension of the quantity.
Just as in the Principia Newton had begun by finding the moment
of the product AB, so Leibniz determined the "difference" of the
product xy. Although at the outset Leibniz was uncertain about his
method and hesitated as to whether or not d{xy) is the same as dxdy
( X \ Q/OC
and whether d[-\ is equal to — , 56 he in the end answered these
\yj dy
questions correctly, determining that d(xy) = xdy + ydx and dl — \
'vdoc "—* ocd/v
= -. These values he found by allowing x and y to become
y 2
x + dx and y + dy respectively. Upon subtracting the original value of
the function from the new one and observing that dxdy is infinitely small
in comparison with the terms xdy and ydx, the results are obtained.
Having established the rules for differences and quotients, Leibniz
was then able to extend these to all integral powers of a variable, the
difference of x n being nx n ~ l dx. Because summation is the inverse of
determining a difference, the integral of x is, of course, ."
n + 1
We have had occasion to observe the derivation of the latter result in
various forms in the methods of previous investigators, but only in
the work of Newton and Leibniz was it obtained as the inverse of
another fundamental operation. A comparison of the derivations by
M See James Bernoulli, "Analysis problematis antehac propositi," p. 218.
66 Leibniz, Early Mathematical Manuscripts, p. 102; Gerhardt, Die Entdeckung der
Differentialrechnung durch Leibniz, pp. 24, 38.
67 Cf. Mathematische Schriften, V, 226 ff.
206 Newton and Leibniz
the methods of fluxions and differences with those by quadratures as
given by Cavalieri, Torricelli, Roberval, Pascal, Fermat, and Wallis
will convince one of the enormous operational facility to be gained by
such a method of procedure. As Newton made the rules for fluxions
basic in his method, so also Leibniz looked upon the operation of
finding "differences" as fundamental in his "differential and sum-
matory" calculus. Ever since Newton and Leibniz invented their
methods and combined them in this manner with the discovery of the
fundamental inverse property, this point of view has continued in
the elementary calculus. Differentiation is in general the fundamental
operation, integration being regarded simply as the inverse of this.
There has been retained also in the calculus a certain element of
confusion in terminology which is the result of the somewhat different
attitudes of Newton and Leibniz with respect, not to the determination
of the integral, but to its definition. Newton defined the fluent as the
quantity generated by a given fluxion — that is, as the quantity having
a given magnitude as its fluxion, or as the inverse of the fluxion. Ir
keeping with this emphasis upon the indefinite integral, Newton
included in both the Methodus fluxionum and De guadratura what
amounts to a table of integrals. Leibniz, on the other hand, defined
the integral as the sum of all the values of a magnitude, 58 or the sum
of an infinite number of infinitely narrow rectangles, or — as modern
mathematics would express it — as the limit of a certain characteristic
sum. These two points of view have been perpetuated in the elementary
calculus, in which there are two integrals: the indefinite integral and
the definite integral. The origins of these, in the history of the subject,
are even now sometimes brought vividly to mind in referring to the
former as "the integral in the sense of Newton" and to the latter as
"the integral in the sense of Leibniz." 69 Too much stress should not
be put upon such a distinction, however, because Newton and Leibniz
were both well aware of the two aspects of the integral. 60
68 "Seu data differentia dy invenire terminum y, est invenire summas omnium differ-
entiarum seu dy." See Leibniz, "Isaaci Newtoni tractatus duo." Cf. also Gerhardt, Die
Entdeckung der Dijjerentialtechnung dutch Leibniz, p. 45.
69 Cf. Saks, Theotie de VinUgrale, p. 122.
80 Hoppe ("Zur Geschichte der Infinitesimalrechnung," pp. 186-87), misled by Newton's
symbol o, has completely misinterpreted the facts on this point. He would have Newton
make the summation of indivisibles fundamental and have Leibniz oppose this point of
view by emphasizing the operation of differentiation. The situation is, on the contrary, if
anything quite the reverse of this.
Newton and Leibniz 207
Newton and Leibniz are known as the founders of the calculus largely
because they established, in the periods 1665-66 and 1673-76 respec-
tively, the methods and relationships outlined above. There was an-
other aspect of their work which the inventors felt carried great
weight — the generality of their methods. Both men pointed out that,
unlike the anticipatory procedures, their methods were applicable
even in the case of radicals. The justification for such an assertion
was made by Newton largely upon the basis of infinite series. If
(x + o) n is to be expanded by the binomial theorem, the number of
terms will be infinite for values of n which are not positive integers.
No conclusion can in general be drawn from an application of the
theorem in this case unless the series is convergent, but neither Newton
nor his successors for a century later fully appreciated the need for
investigations into the question of convergence. Leibniz in this respect
had perhaps even less caution than many of his contemporaries, for
he seriously considered whether the infinite series 1 — 1 + 1 — 1
+ 1 — . . . was equal to ^. G1 He had as well fewer scruples than Wallis
in the wide generalization of rules demonstrated only for a small
number of special cases. Although he had indicated a proof of his
rule for the "difference" of x n for integral values of n only, he an-
nounced that this would hold for all values, and entitled the first
printed account of the calculus "A New Method for Maxima and
Minima, as Well as Tangents, Which Is Not Obstructed by Fractional
or Irrational Quantities." 62
This first published treatise of the calculus, a memoir of six pages
appearing in the Acta erudilorum of 1684, three years earlier than
Newton's first account, 63 must have repelled most readers seeking an
introduction to the new method. Even to the Bernoulli brothers, who
did so much to popularize the subject in its early stages, it was "an
enigma rather than an explication." 64 In the first place, it contained
many misprints. 65 Secondly, Leibniz here imitated the bareness and
61 Cf. Mathematische Schriften, V, 382-87.
62 "Nova methodus pro maximis et minimis, itemque tangentibus, que nee fractas nee
irrationales quantitates moratur, et singulare pro illis calculi genus." Acta erudilorum,
1684, pp. 467 ff. See Mathematische Schriften, V, 220 ff. For an English translation of this
see Raphson, History of Fluxions, pp. 19-27.
03 See Matliematische Schriften, V, 220 ff .
64 See Leibniz, Matliematische Schriften, III (Part 1), 5, n.
65 Enestrom, "Uber die erste Aufnahme der Leibnizschen Differentialrechnung."
208 Newton and Leibniz
simplicity of the Geometrie of Descartes, although in later articles he
attempted fuller explanations. His 1684 work contained, besides the
definition of the "difference" or differential of a quantity, the rules —
without proof — for the differentials of sums, products, quotients,
powers, and roots, with a few applications to tangents and to prob-
lems in maxima and minima and points of inflection. It is interesting
that among these examples is one in which Leibniz derived the law
of refraction, using Fermat's principle. This suggests the influence of
Fermat, who had himself given a similar demonstration and whose
work was repeated later by Huygens; but Leibniz did not refer to
him here. Although quadratures were not included in this first paper,
Leibniz developed the applications to such problems two years later,
in another paper in the Acta eruditorum.™ In subsequent articles in
this and other journals, 67 Leibniz gave further developments and
applications of his calculus — such as the determination of the differ-
entials of logarithms and exponentials, and of osculating figures.
In all of this work Leibniz realized that he was creating a new
subject. It has been suggested that it was only after the method of
Leibniz had achieved marked success that Newton came to regard
the method of fluxions as constituting a new subject and an organized
mode of mathematical expression, 68 rather than simply as a useful
modification of some earlier rules. This is belied by the fact that
Newton had by 1676 written out three accounts of his method; but it
remains true, nevertheless, that Leibniz expressed himself more vig-
orously on this point than did Newton. He said that his analysis was
to be compared with the methods of Archimedes in much the way
that the work of Viete and Descartes had been to the geometry of
Euclid, 69 in that it dispensed with the necessity of imagination. In
order to popularize it, he announced explicitly all the rules of opera-
tion, even the simplest, 70 presenting these as though they were rules
of algebra, and pointing to the reciprocal relation of powers and
roots as analogous to that subsisting between his "sums" and "dif-
ferences," or integrals and differentials. 71
66 "De geometria recondita et analysi indivisibilium atque infinitorum." Acta erudito-
rum, 1686, pp. 292-300; see also Leibniz, Mathematische Schriften, V, 226 ff.
67 See Leibniz, Mathematische Schriften, V, for these papers.
68 De Morgan, Essays on the Life and Work of Newton, pp. 32-34.
69 Mathematische Schriften, II, 123.
70 Zeuthen, "Notes,"' 1895, p. 236. 71 Mathematische Schriften, V, 231, 308.
Newton and Leibniz 209
This didactic spirit of Leibniz is in contrast with the reticence which
Newton displayed on the subject of his method of fluxions, perhaps
from a morbid fear of opposition. With reference to the logical and
philosophical justifications of his procedures, on the other hand,
Leibniz was less emphatic than Newton. He did not make a really
serious effort on this point, because he felt that the calculus, as a
modus operandi, brought its demonstrations with it. He did not wish
to make of the infinitely small a mystery, as had Pascal; nor did he
turn to geometrical intuition for clarification. In appealing only to
intelligence, he stressed rather the algorithmic nature of the method,
as he himself spoke of it. In this sense he may justly be considered one
of the founders of formalism, as opposed to intuitionism, in mathe-
matics. He had confidence that if he formulated clearly the appro-
priate rules of operation and that if these were properly applied, some-
thing reasonable and correct would result, 72 howsoever doubtful might
be the meaning of the symbols involved. This attitude reflects well the
corresponding difficulties experienced at the time with imaginary
numbers. Leibniz, unlike Aristotle, seemed to feel that his position
was to be justified by an appeal to the principle of sufficient reason to
determine, in this connection, the transition from possibility to
actuality. 73 The importunateness of his contemporaries, however, made
it necessary for him now and then to attempt a further clarification
of the basic conceptions of his differential calculus. In this respect he
was neither lucid nor consistent. 74
From his earliest to his latest work Leibniz made use of the prin-
ciple that in a relationship containing differentials of various orders,
those of lowest order only are to be retained, because all the others
will be infinitely small with respect to these. This is, in a new algebraic
form, essentially the same as the doctrine which Roberval and Pascal
had employed when they had held that a line is as nothing in com-
parison with a square. Leibniz had carried over into analysis the
idea of infinitely small quantities of different orders, based upon a
principle of homogeneity which geometrical considerations had sug-
gested. Whereas Fermat, Barrow, and Newton had made use only of
72 Klein, Elementary Mathematics from an Advanced Standpoint, p. 215.
73 Enriques, Historic Development of Logic, p. 77.
74 Hoppe ("Zur Geschichte der Infinitesimalrechnung," p. 184) has maintained that
the thought of Leibniz was much deeper and more accurate than that of Newton, but his
view is based upon a misinterpretation of the work of the latter.
210 Newton and Leibniz
first order infinitesimals, Leibniz conceived of an infinite number of
such orders, corresponding in a sense to the infinite ranks in the
system of monads found in his philosophical scheme. However, in his
definition of the differential of the first order, Leibniz vacillated; and
for those of higher order he was very far from giving a satisfactory
explanation.
In the first published account of the calculus, Leibniz gave a
singularly satisfactory de finiti on of his first-order differentials. He
said that the differential dx of the abscissa x is an arbitrary quantity,
and that the differential dy of the ordinate y is defined as the quantity
which is to dx as the ratio of the ordinate to the subtangent. Barrow's
rule for tangents had, in a sense, implied a similar definition, for it
had required the substitution of the ordinate for a and the subtangent
for e; but Barrow's a and e were vague infinitesimals. In the definition
of Leibniz given above, the differentials are finite, assignable quantities,
entirely comparable with those defined in the calculus of today. This
fact has led to the assertion that "Leibniz from the beginning of the
new calculus defined the differential absolutely as did Cauchy." 75
In a sense this is true, but such a statement is quite misleading for
two reasons. In the first place, the definition of Leibniz presupposes
logically a satisfactory definition of the tangent line, just as Cauchy's
differential depends upon the notion of the derivative. In each case it
would be expected that the explanation should be in terms of limits.
Leibniz, however, unlike Cauchy, defined the tangent as a line joining
two infinitely near points of the curve, these infinitely small distances
being expressible by means of differentials or differences between two
consecutive values of the variable. 76 This constitutes a petitio principii
which indicates that the avoidance of infinitely small quantities in the
thought of Leibniz was only superficial. Of course, it is possible that
Leibniz intended his language to be interpreted in the precise sense
of limits, as we do when we speak of the tangent as the line through
two consecutive or coincident points; but a further consideration of
Leibniz's writings will make such a view appear a misinterpretation
of his whole thought. Leibniz, throughout his work, regarded the
n Mansion, Resume du cours <T analyse infinite" simale, Appendix, "Esquisse historique,' ?
p. 221; cf. also editorial note, Malhesis, IV (1884), 177.
78 Mathematiscke Schriften, V, 220 ff.; Gerbardt, Die Entdeckung der Differentialrechnung
dutch Leibniz, p. 35.
Newton and Leibniz 211
differential as fundamental, as a recent study has shown. 77 Modern
mathematics agrees with Cauchy, however, in making this notion
subordinate to that of a limit by denning it in terms of the derivative.
The reason for this change of view is to be found in the failure of
attempts made by Leibniz and others to give a satisfactory definition
of the differential independent of the method of limits.
The attempts of Leibniz to give satisfactory definitions of differ-
entials of higher orders were unsuccessful. He said that ddx or d 2 x
is to dx as dx is to x™ making no distinction between the differentials
of independent and dependent variables. Somewhat similarly he said
that if dx : x = dh : a, where a is a constant and dh is a constant
differential, then d 2 x : dx = dh : a or d 2 x : x - dh 2 : a 2 ; and in general
d e x : x = dti : a e , where e may even be fractional. 79 Perhaps realizing
that this definition could not be consistently applied, he later gave a
geometrical interpretation which, although lacking precision of state-
ment, can be correctly interpreted in terms of derivatives. Given any
curve, let dx be at every point an assignable quantity and let dy be
such that the ratio of dy to dx is that of the ordinate to the subtangent.
If then for every point on the curve we plot on the same axes a new
point whose ordinate is proportional to dy, the result will be a new
curve, the differentials of which will be the "differentio-differentials,"
or "second differentials," of the original curve. 80 This geometric rep-
resentation is in general equivalent to that which would be obtained
dy
by plotting the values of the ratio — for every point. The second
dx
differential d 2 y would then be determined from the derivative of the
new curve — that is, from the second derivative of the original curve.
Leibniz, however, did not regard derivatives as fundamental, so that
his remarks here cannot be regarded as constituting a satisfactory
definition of d 2 y, any more than can that of dy in terms of the tangent.
The lack of such suitable definitions led to bizarre uses of the differ-
ential symbolism. In 1695 John Bernoulli, in a letter to Leibniz,
3 / — d 3 y
wrote such expressions as: V d 6 y = d 2 y and — = d?yd~ 2 x = d 3 ypx. m
d 2 x
77 Petronievics, "tJber Leibnizens Methode der direkten Differentiation."
78 Mathematische Schriften, V, 325; cf. also III (Part 1), 228.
79 Ibid., Ill (Part 1), 228. » "Addenda ad schediasma proximo," p. 370.
« Leibniz, Mathematische Schriften, III (Part 1), 180.
212 Newton and Leibniz
In the absence of satisfactory definitions, Leibniz resorted fre-
quently to analogies to clarify the nature of his infinitely small dif-
ferentials. At one point he made use of the imagery of Newton and
spoke of his differentials as the momentary increments or decrements
of quantities. 82 Again he applied the thought of Hobbes and said that
the conatus is to motion as a point to space or as one to infinity. 83
The infinitely small he considered the study of the vanishing or incip-
iency of magnitudes, as opposed to quantities already formed. 84 He
applied this analogy to second-order differentials as well. If in nature
a motion is thought of as pictured by a line, then the impetus or
velocity is represented by an infinitely small line, and the acceleration
by a line doubly infinitely small. 86
Appealing again to geometrical intuition, Leibniz said that as a
point added nothing to a line, because it is not homogeneous or com-
parable, so differentials of a higher order in his method may likewise
be neglected. 86 Pursuing this thought further, he said that if one
thinks of geometrical magnitudes as represented by the ordinary quan-
tities of algebra, then the first differentials refer to tangents, or the
directions of lines, and the higher differentials refer to osculations or
curvatures. 87 In this sense the differentials were to be compared with
the Euclidean angle of contact which was less than any assignable
magnitude and yet not zero. 88
In the absence of rigorous definitions Leibniz continued to multiply
analogies. In a somewhat less critical vein he said that the differential
of a quantity can be thought of as bearing to the quantity itself a
relationship analogous to that of a point to the earth or of the radius
of the earth to that of the heavens. 89 In another place he said that
as the earth is infinite with respect to a ball held in the hand, so the
distance of the fixed stars is doubly infinite with respect to the ball; 90
and this analogy he repeated later, substituting a grain of sand for the
ball.
John Bernoulli, the pupil of Leibniz, with great naivet6 pointed
out other analogies which the work of Galileo and of Leeuwenhoek
M Mathematische Sckriften, VH, 222.
M Pkilosophische Sckriften, IV, 229. Cf . also Mathematische Schrijten, III, 536 ff.
M Pkilosophische Sckriften, VI, 90. " "Testamen de motuum coelestium," p. 86.
» Mathematische Sckriften, V, 322. * Ibid., V, 325-26; cf. also p. 408.
88 Mathematische Sckriften, V, 388. a "Testamen de motuum coelestium," p. 85.
*> Mathematische Sckriften, V, 350, 389.
Newton and Leibniz 213
had made possible. He compared the orders of the infinitely great with
the relations of the stars to the sun, to the planets, to the satellites of
these, to the mountains on the last named, and so on. The infinitely
small quantities in the same way resembled the apparently number-
less grades of animalcules which the microscope had disclosed. 91
Leibniz, while appreciating the comparison, cautioned Bernoulli that
the latter were of finite size, whereas the differentials were infinites-
imals. 92
It is interesting to notice that whereas Newton had used the
infinitesimal conception in his early work, only to disavow it unequiv-
ocally later and to attempt to establish the idea of fluxions on the
doctrine of prime and ultimate ratios of finite differences — that is,
in terms of limits — we shall find that with Leibniz the tendency is
somewhat in the other direction. Beginning with finite differences, he
was to be confirmed in his use of infinitesimal conceptions by the
operational success with which his differential method met, although
he seems to have remained largely in doubt as to its logical justifica-
tion. The divergent views of these two men were perhaps less the result
of dissimilar mathematical traditions than of varying tastes. Newton,
the scientist, found in the notion of velocity a basis which to him
appeared to be satisfactory; Leibniz, the philosopher, who was, as
well, perhaps as much a theologian as a scientist, 93 preferred to find
this in the differential, the counterpart in thought of the monad,
which was to play such a large part in his metaphysical system.
Although Leibniz continued to employ the conceptions and the
methods of infinitesimals, his justification of the infinitely small was
not at first the result of serious effort, inasmuch as the notion had
been employed by geometers with more or less indulgence throughout
the century. Although Huygens and some few others did not readily
accept the new calculus, they had not opposed it. In 1694, however,
the Dutch physician and geometer, Bernard Nieuwentijdt, who was
in this respect to be succeeded by a long line of mathematicians,
opened an attack upon the lack of clarity in Newton's work and
upon the validity of the higher differentials of Leibniz. Although he
admitted in general the correctness of the results of the new methods,
91 Leibniz and Bernoulli, Cotnmercium philosophicum et ntathematicum, I, 410 ff.
<* Ibid.; cf. also Mathematische SckrifUn, HI (Part 2), 518, 524.
93 Mach, Science of Mechanics, p. 449.
214 Newton and Leibniz
he felt that they entailed some obscurity and said that they not infre-
quently led to absurdities. The tangent procedure of Barrow he
criticized because a and e were taken as zero. 94 Newton's evanescent
quantities he regarded as too vague, and he said that he could not
follow the reasoning in the lemmas on limits, by which quantities,
which in a given time tend to equality, are in the end equal. 95 In the
analysis of Leibniz he questioned the manner in which the sum of
infinitesimals might be a finite quantity. 96
In a further attack in the following year Nieuwentijdt said that
Leibniz had not made the nature of infinitely small quantities any
clearer than had Newton and Barrow, nor could he explain in what
way the differentials of higher order differed from those of the first
order. 97 Nieuwentijdt tried to develop a method which would solve
the problems of Leibniz without using infinitesimals of higher order;
but the result was a failure. 98 The infinite and infinitesimal Nieu-
wentijdt defined unsatisfactorily, as had Nicholas of Cusa, as quan-
tities respectively greater than and less than any given magnitude. 99
Thus, whereas the work of Newton and Leibniz had tended, in certain
respects, toward the limit concept, the view of Nieuwentijdt repre-
sented a reversion to a less critical manipulation of the static infinite
and the infinitesimal, which in the following century was boldly
extended to the infinitely large and the infinitely small of higher
orders, particularly by Fontenelle. This was unfortunate, as a serious
effort to establish a secure foundation for the calculus was to be desired
at the time.
Leibniz answered Nieuwentijdt in 1695 in the Acta eruditorum, 100
defending himself from "overprecise" critics, whom he likened to the
Skeptics of long before. He maintained that we should not be led by
excessive scrupulousness to reject the fruits of invention. His method,
he maintained, differed from that of Archimedes only in the ex-
pressions used, being in this respect more direct and better adapted
to the art of discovery. 101 After all, he held, the phrases "infinite''
94 Nieuwentijdt, Consideraiiones circa analyseos ad quantitates infinite parvas applicatae
f/rincipia, p. 8
98 Ibid., pp. 9-15. »« Ibid., p. 34; cf. also pp. 15-24.
97 Analysis infinitorum seu curvilineorutn proprietates ex polygonorum natura deduclae.
98 See Weissenborn, Die Principien der hoheren Analysis, Section 10.
99 Analysis infinitorum, p. 1.
190 See Mathematische Sckriften, V, 318 ff. m Mathematische Sckriften, V, 350.
Newton and Leibniz 215
and "infinitesimal" "signify nothing but quantities which one can
take as great or as small as one wishes" — somewhat as Aristotle had
regarded the infinitely small as indicating a potentiality only — "in
order to show that an error is less than any which can be assigned —
that is, that there is no error." 102 Nevertheless, he reiterated that one
can use these as "ultimate things" — that is, as actually infinite and
infinitely small quantities — as a tool, much "as algebraists retain
imaginary roots with great profit." 103 This Janus-faced appearance of
the differential recurs frequently in the work of Leibniz. Although he
commonly spoke of it as infinitely small, he often used instead the term
incomparably small. 104 He held that "if one preferred to reject infi-
nitely small quantities, it was possible instead to assume them to be
as small as one judges necessary in order that they should be incom-
parable and that the error produced should be of no consequence, or
less than any given magnitude." 105
At another point Leibniz said that the differential was "less than
any given quantity" and compared the neglect of differentials to the
work of Archimedes, who "assumed, together with all those following
him, that quantities which did not differ by a given quantity were in
fact equal." 106 He here seemed to feel that his method constituted but
a cryptic form of expression of the method of exhaustion, so that
further justification was superfluous. However, the meaning of the
symbols themselves was still not clear.
The hesitation which Leibniz displayed as to whether the differ-
entials were to be regarded as assignables or inassignables is perhaps
most neatly illustrated by a procedure he suggested in his Historia et
origo calculi differentialis, written a year or two before his death. In
order to avoid working with quantities which are not really infini-
tesimals but which are treated as such — amphibia between existence
and nonexistence, as Leibniz called the imaginary numbers — he re-
sorted to a bit of sleight of hand. Leibniz here used the symbols
{d)x and (d)y to represent finite assignable differences; then, after the
calculation had been completed, he replaced them by the inassignable
m PhUosophische Schriften, VI, 90.
108 Leibniz, Early Mathematical Manuscripts, p. 150.
104 Mathematische Schriften, V, 407; cf. also p. 322.
106 "Testamen de motuum coelestium > ,, p. 85.
"* "Responsio ad nonnullas difficultates, a Dn. Bernardo Nieuwentijt," p. 311.
216 Newton and Leibniz
infinitesimals or differentials dx and dy, "as a kind of fiction," because,
after all, "dy : dx can always be reduced to a ratio (d)y : d(x) between
quantities which are without any doubt real and assignable." 107
Just how this jump from assignables to inassignables and back
again is to be justified, Leibniz did not make plain. It appears from
this argument that he realized that it was not the individual differ-
entials, but only their ratio, that constituted the important considera-
tion. Newton likewise understood that the significance of his method
lay in the ratio of fluxions, so that for fluxions one could substitute
other finite quantities in the same proportion. That the limit concept
does not stand out clearly in their work is probably the result of the
fact that they, and their contemporaries, were thinking always of a
ratio as the quotient of two numbers, rather than as a single number
in its own right. 108 Only after the development of the general abstract
concept of real number was the way clear to interpret both the
fluxionary and the differential calculus in terms of the limit of an
infinite sequence of ratios or numbers; but this interpretation did not
become generally acceptable for another century. The meanings of
the terms "evanescent quantities" and "prime and ultimate ratio"
had not been clearly explained by Newton, his answers being equiv-
alent to tautologies:
But the answer is easy: for by the ultimate velocity is meant that, with
which the body is moved, neither before it arrives at its last place, when
the motion ceases, nor after; but at the very instant when it arrives. . . .
And, in like manner, by the ultimate ratio of evanescent quantities is to be
understood the ratio of the quantities, not before they vanish, nor after,
but that with which they vanish. 109
This sounds very much like the quotient of two infinitesimals, al-
though Newton a,dded that "those ultimate ratios with which such
quantities vanish are not truly the ratios of ultimate quantities but
the limits to which the ratios of quantities, decreasing without end,
always converge." In other words, it is the ratio in which Newton
was interested, not the evanescent quantities themselves; but he failed
to define this ratio unequivocally.
107 See Leibniz, Early Mathematical Manuscripts, p. 155; Weissenborn, Die Principien
der hoheren Analysis, p. 104; Gerhardt, Die Entdeckung der Differentialrechnung durch
Leibniz, p. 31.
108 Pringsheim, "Nombres irrationnels et notion de limite," pp. 143-44.
108 Opera omnia, I, 250-51; II, 40-41.
Newton and Leibniz 217
Leibniz had a somewhat similar idea. Just as Newton never cal-
culated a single fluxion but always a ratio, so also Leibniz realized
that it was the ratio of, or relationship between, the differentials
which was significant. These could therefore be regarded as any finite
quantities, the ratio of which was that of the ordinate to the sub-
tangent. For pragmatic reasons, however, Leibniz retained the infi-
nitely small, justifying this by saying that if one desired rigor, he could
for the "inassignables" substitute "assignables" having the same
ratio. However, just as Newton did not clearly explain how ratios of
evanescent quantities become, or are related to, "prime" or "ultimate"
ratios, so Leibniz was unable to explain the transition from finite to
infinitesimal magnitudes. Leibniz admitted that one could not prove
or disprove the existence of infinitely small quantities. 110
Furthermore, Leibniz felt that the justification for his calculus lay
in the ordinary mathematical considerations already known and used,
and that it was "not necessary to fall back upon metaphysical contro-
versies such as the composition of the continuum." 111 Nevertheless,
when called upon to explain the transition from finite to infinitesimal
magnitudes, he resorted to a quasi-philosophical principle known as
the law of continuity. We have seen previous applications made of
this doctrine by Kepler and by Nicholas of Cusa. The latter may have
influenced Leibniz in this respect, as well as in the philosophical
doctrine of monads. 112
Leibniz, however, gave to the doctrine of continuity a clarity of
formulation which had previously been lacking and perhaps for this
reason looked upon it as his own discovery. This "postulate" Leibniz
expressed, in a letter to Bayle in 1687, as follows: "In any supposed
transition, ending in any terminus, it is permissible to institute a
general reasoning, in which the final terminus may also be included." 113
Thus in his manipulations in the calculus, "the difference is not
assumed to be zero until the calculation is purged as far as is possible
by legitimate omissions, and reduced to ratios of nonevanescent
quantities, and we finally come to the point where we apply our result
m Leibniz and Bernoulli, Commercium philosophicum et tnathematicum, I, 402 ff.; cf. also
Mathematische Schriften, III (Part 2), 524 ff.
m Leibniz, Early Mathematical Manuscripts, pp. 149-50.
m See Zimmermann, "Der Cardinal Nicolaus Cusanus als Vorlaufer Leibnizens."
m Leibniz, Early Mathematical Manuscripts, p. 147; see also Philosophische Schriften,
TJl y 52; Mathematische Schriften, V, 385.
218 Newton and Leibniz
to the ultimate case" 114 — ostensibly by virtue of the law of continuity.
Thus even in the work of Leibniz the idea of a limit was implicitly
invoked, although the logical order was reversed. Leibniz justified the
limiting condition by the law of continuity, whereas mathematics has
since shown that the latter must itself first be defined in terms of
limits. In this manner of thinking Leibniz seems still to be striving to
make use of a vague idea of continuity which we feel we possess and
which had bothered thinkers since the Greek period.
Newton, of course, had tacitly ensconced part of his difficulty with
the continuum in the comforting notion of continuous motion, al-
though he too, in his prime and ultimate ratio, was implicitly invoking
the law of continuity of Leibniz. While Newton sought to avoid the
limit concept by notions acceptable to scientific empiricism, Leibniz
had recourse to ideas of ultimate form, suggested by metaphysical
idealism. Even when the quantities — such as differentials — involved in
a relationship become inassignable, nevertheless he felt that the ulti-
mate form remained. A point, he held, was not that whose part is
zero, but whose extension is zero. 115 In a similar connection Leibniz
had asked Wallis, "Who does not admit a figure without magnitude?" 116
The characteristic triangle was for Leibniz one in which the form of
the triangle remained after all magnitude had been abstracted from
it, 117 much as Newton had spoken of the ultimate forms of evanescent
triangles. 118
Leibniz had intended to write a volume on the subject of the
infinite. This book, which would have constituted a definitive ex-
pression of his views, did not appear. However, the attitude which
Leibniz expressed in connection with his law of continuity seems to
have been that which, with some vacillations and modifications, he
held until his death. In his Theodicee he had said, in speaking of
infinite and infinitely small quantities, "but all that is nothing but
fictions; every number is finite and assignable, as is every line like-
wise." 119 Nevertheless, several years later, in writing to Grandi, he
said, "Meanwhile, we conceive the infinitely small not as a simple
n * Leibniz, Early Mathematical Manuscripts, pp. 151-52.
U5 PkUosophische Schriften, IV, 229.
u « Maihematische Schriften, IV, 54; cf. also IV, 63.
w Cf. Freyer, Studien zur Metaphysik, p. 10.
m Opera omnia, I, 243. 1W Philosophische Schriften, VI, 90.
Newton and Leibniz 219
and absolute zero, but as a relative zero, (as you yourself well remark),
that is, as an evanescent quantity which yet retains the character of
that which is disappearing." 120
Leibniz here clearly has in mind the law of continuity. This prin-
ciple, the origin of which lay in the nature of the infinite, he felt to be
absolutely necessary in geometry, as well as useful in physics. 121
Accordingly, he regarded an equality as a particular case of an
inequality, and an infinitely small inequality as becoming an equality. 122
Acceptance of the law of continuity would therefore justify the
omission of differentials of higher order, and it appears that it is upon
this basis that Leibniz would have his calculus justified. A rather
inexact tradition would impute to Leibniz a belief in actually infin-
itesimal magnitudes. 123 However, Leibniz himself, in a letter written
about two months before his death, said emphatically that he "did not
believe at all that there are magnitudes truly infinite or truly infinites-
imal." 124 These conceptions he regarded as "fictions useful to abbre-
viate and to speak universally." 125 The link between these fictions and
reality he undoubtedly felt would be found in his law of continuity,
which he had taken as basic in all his later work in the calculus. 126
We have traced the development of the calculus through the
invention of the methods of Newton and Leibniz, and have found that
the concepts involved had not yet been clarified. Newton gave three
interpretations of his procedure, and although indicating a preference
for the notion of prime and ultimate ratios as the most rigorous, he
did not elaborate any one into a careful logical system. Leibniz dis-
played a similar lack of decision, for although he employed the
infinitesimal method throughout, he wavered in his attitude toward
differentials, considering them variously as inassignables, as quali-
tative zeros, and as auxiliary variables.
It has very misleadingly been said that Leibniz developed his method
logically, while Newton was changeable in his conceptions, using
limiting ratios to mask his infinitesimals. 127 The converse can, with
120 Malhematische Schriften, IV, 218 f.
121 Philosophised Schriften, III, 52. m Ibid., II, 105.
m Cf. Klein, Elementary Mathematics from an Advanced Standpoint, p. 214.
™ Opera omnia (Dutens), HI, 500. ™Ibid.
m Scholtz, Die exakte Grundlegung der Infinitesimalrechnung bei Leibniz, p. 39.
i* 7 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 184; Moritz Cantor, "Origines
du calcul infinitesimal," p. 24.
220 Newton and Leibniz
perhaps more justice, be asserted. The position of Leibniz was more
difficult to maintain than had been that of the users of indivisibles,
for he was required to explain infinitesimals not only of first order, but
of all orders. Newton had avoided a multiplicity of infinitesimals
through the method of fluxions, which required only a single incre-
ment — that in the time, o. His method was consequently well adapted
to an interpretation in terms of limits, assuming that the idea of
fluxion has first been properly defined or is recognized as a primary,
undefined notion. Leibniz, on the other hand, had no quantity to
serve as an independent variable, so that although he recognized that
it was not the single differentials but only their ratio which was
significant, he could not indicate how this fact was to be applied in his
work. Newton had experienced this same difficulty when, in his Principia
demonstration, he abandoned the fluxionary method and resorted to a
use of infinitesimals in connection with a function of two variables.
Furthermore, the notation of Leibniz concealed, perhaps more
effectively than that of Newton, the logical basis of the calculus.
Leibniz had developed — as the result of painstaking investigation,
patient experimentation, and frequent correspondence on the subject
with other mathematicians 128 (particularly the Bernoulli brothers) —
a symbolism which was remarkably felicitous when applied to the
solution of problems. Because it is so convenient as to be almost
automatic, this notation has been maintained to the present day.
Nevertheless, this very success operated to mislead Leibniz as to the
rigorous formulation of the subject. His system of notation caused him
to think of his differential ratios as quotients, and his integrals as
sums — of what, he could not say — rather than as limiting values of
certain characteristic functions. His view seems to have been that
first-order differentials were incomparables (a useful fiction), which
by the law of continuity had the same ratio as that of the ordinate
to the subtangent. Second-order differentials were the inassignables
which were to those of first order as the latter were to unity. Such a
j /- • • d 2 y (dy\ 2
definition of course confused — with I — 1 , and this unsatisfactory
0-x \CLdC I
state of affairs was not cleared up until the infinitesimal calculus was
explained later in terms of limits.
128 Cajori, A History of Mathematical Notations, II, 180-81.
Newton and Leibniz 221
Newton's notation likewise offered difficulties in the way of logical
formulation. Not only did the infinitely small insinuate itself into the
method of fluxions in the form of a "moment," but Newton occasion-
ally failed to exercise care in distinguishing clearly between fluxions
and moments. Furthermore, in the terminology of "prime and ultimate
ratios" there was the idea, which was to persist for a century, that
the limiting quantities were ratios, rather than single numerical values.
Newton recognized that the ultimate ratio was a ratio of fluxions and
not of moments. However, he was not clear in his explanations on
this point and failed to point out the analogy between this ratio and
the so-called sum of an infinite series. As the Euclidean view of
number had been based upon the ratio of geometrical magnitudes,
so here Newton emphasized the ratio of two velocities, rather than the
single limiting value of a function, as brought out in the nineteenth-
century definitions of the derivative.
With such indecision and lack of clarity on the part of the inventors
of the calculus, it is hardly surprising to discover confusion among
their followers as to the nature of the subject. This was intensified
by the fact that many mathematicians failed completely to distinguish
between the two systems and so misunderstood the arguments of
their authors. This lack of distinction was due to some extent to
Newton and Leibniz themselves. In the Acta eruditorum of 1705 129 an
account of Newton's De guadratura was given, probably by Leibniz,
in which it was stated that in this book the differentials of Leibniz
were merely replaced by fluxions, and it is implied that these are
essentially the same. In England, the committee of the Royal Society
investigating the priority claims of Newton and Leibniz reported in
the Commercium epistolicum of 17 12 130 that "The differential metliod
is one and the same with the method of fluxions, excepting the name
and mode of notation; Mr. Leibniz calling those quantities differences
which Mr. Newton calls moments or fluxions, and marking them with
the letter d, a mark not used by Mr. Newton." 131 Thus even Newton's
own countrymen did not realize that although he had used infinitely
m See Leibniz, "Isaaci Newtoni tractatus duo," p. 34; cf. also Bertrand, "De l'invention
du calcul infinitesimal."
1,0 See Newton, Opera omnia, IV, 497-592, for a reprint of this. See also the review
(probably by Newton), Philosophical Transactions, XXIX (1714-16), pp. 173-224.
m See Opera omnia, IV, 588-89.
222 Newton and Leibniz
small moments in his early work, he professed to have nothing to do
with them in his later expositions in terms of fluxions and of prime
and ultimate ratios. This is more easily understood when we recognize
that Newton himself never admitted a change in his view, and that
his prime and ultimate ratio, unless interpreted under the limit
concept or the principle of continuity, will involve the infinitely small.
The first published volume devoted expressly to the history of the
calculus — The History of Fluxions by Joseph Raphson — appeared in
1715 and illustrated well the prevailing confusion of thought. To the
author of this volume and to Halley, Newton had in 1691 planned to
entrust the preparation for publication of his De guadratura. m How-
ever, a change of heart led Newton to withhold publication until
1704, by which time he had, as has been noted, emphatically re-
nounced infinitesimals. Raphson, nevertheless, failed to recognize any
change in the point of view and continued to confuse fluxions and
moments. The small letters, which in the Principia Newton had
employed to designate moments, Raphson interpreted as symbols of
fluxions, and these he identified (all too uncritically) with the infin-
itesimals of Barrow and Nieuwentidjt, and with the differentials of
Leibniz. 133 Raphson went so far as to regard fluents as infinitely great
with respect to the finite quantities from which they were obtained;
and fluents of higher orders were in similar manner interpreted as of
higher degrees of infinity. 134
In all fairness it is to be remarked that the object of Raphson's
work was not to furnish a clarification of the basic notions of the
calculus. The book contains, in fact, no formal definitions for such
terms as fluxion, moment, or fluent. The purpose was partly to present
the various methods in such a way as to make as easy as possible
the application of the rules of procedure to specific problems, but
"chiefly to do justice to their authors in point of chronology." 135 In
both respects Raphson vigorously maintained the superiority of the
work of Newton. With respect to priority his views were essentially
those expressed in the report in the Commercium epistolicum of three
years earlier, suggesting the possibility of plagiarism on the part of
Leibniz. 136
m See Raphson, The History of Fluxions, pp. 2-3. 133 Ibid., p. 4.
"* Ibid., p. 5. «« Ibid. «« Cf. Ibid., pp. 8, 19, 61, 92.
Newton and Leibniz 223
As to the Leibnizian method and notation, Raphson most unjustly
characterized this as "less apt and more laborious" and as a "far-
fetched symbolizing" of "insignificant novelties." 137 This unfortunate
attitude, which excessive deference to the great reputation of Newton
had engendered, was to prevail among British mathematicians for
just about a century following the appearance of Raphson's work,
with the result that the differential notation made little headway in
England before 1816. This period was characterized likewise by the
same confusion of fluxions and the infinitely small as found in the
exposition given by Raphson. As it was, a large number of the textbooks
on the method of fluxions regularly interpreted fluxions as infinitely
small quantities, 138 thus adding to the prevailing confusion in the
interpretation of the conceptual bases of the calculus, which furnished
the provocation for a century of criticism and controversy anent the
subject.
™ Ibid., p. 19.
138 Cajori, A History of the Conceptions of Limits and Fhixions, Chap. II; De Morgan,
"On the Early History of Infinitesimals in England."
VI. The Period of Indecision
THE FOUNDERS of the calculus had clearly stated the rules of
operation which were to be observed, and the astonishing success
of these when applied to mathematical and scientific problems by
Euler, Lagrange, Laplace, and a host of others led men to overlook
somewhat the highly unsatisfactory state of the logic and philosophy
of the subject. Throughout the whole of the eighteenth century there
was general doubt as to the nature of the foundations of the methods
of fluxions and the differential calculus. In England Newton's lack of
clarity and his inconsistency in notation was followed by a confusion
of fluxions with moments. On the Continent the metaphysical ration-
alism of Leibniz was neglected by his followers, who freely attempted
to interpret the differentials as actual infinitesimals or even as zeros,
and who criticized Leibniz for his hesitancy in this respect.
Such a state of affairs could not long continue unchallenged.
Nieuwentijdt had earlier questioned the validity of differentials of
higher orders. Gassendi had referred to the vanity of mathematical
demonstrations based upon the infinitesimal methods, 1 and Bayle had
used the difficulties of the infinite in the service of a general skepti-
cism. 2 However, the most general and significant attack upon the struc-
ture of the new analysis was launched in 1734 by the philosopher and
divine, George Berkeley, in a tract called The Analyst. 3
Berkeley had previously attacked Newton's cosmology in his
Essay towards a New Theory of Vision, but the motive prompting his
animadversions in The Analyst was as largely that of supplying an
apology for theology as it was of inflicting upon the proponents of the
new calculus a rebuke for the weak foundations of the subject. This
we gather from the subtitle of the tract: Or a Discourse Addressed to
an Infidel Mathematician. [This referred to Newton's friend, Edmund
Halley.] Wherein It Is Examined Whether the Object, Principles, and
Inferences of the Modern Anaylsis Are More Distinctly Conceived, or
1 See Bayle, Dictionnaire kistorique et critique, XV, 63.
2 Cohn, Geschichte des Unendlichkeits problems, pp. 193-97.
3 The Works of George Berkeley, Vol. III.
The Period of Indecision 225
More Evidently Deduced, than Religious Mysteries and Points of Faith.
"First Cast the Beam Out of Thine Own Eye; and Then Shalt Thou See
Clearly to Cast Out the Mote Out of Thy Brother's Eye."
Berkeley in this work did not deny the utility of the new devices
nor the validity of the results obtained. He merely asserted, with some
show of justice, that mathematicians had given no legitimate argu-
ment for their procedure, having used inductive instead of deductive
reasoning. 4 His objections were not to the mathematician as an artist
and computist, but as "a man of science and demonstration" and
"in so far forth as he reasons." After giving an account of Newton's
method of fluxions which is eminently fair, except for the irrelevant
ridicule of successive fluxions, Berkeley pointed out his specific
objections. Since Newton had admitted to no change in his point of
view, Berkeley legitimately took advantage of this fact to criticize a
demonstration in the Principia in which the author had made use of
infinitely small quantities in determining the moment of a product. 5
Newton had found the moment of the rectangle AB by giving to
A and B the decrements and increments \a and \b, then subtracting
the diminished rectangle from the augmented. Berkeley objected that
we must take the whole amounts a and b as increments or decrements
to find the moment of AB, and in this case we would have to omit
from the final calculation the infinitely small quantity ab. Berkeley
with perfect right criticized leaving out the infinitely small quantity
ab, if the result is to be rigorously correct, and cited a passage in Be
quadratura, in which Newton professed not to neglect anything, no
matter how small.
Newton had not gone into the justification of his demonstration,
but had indicated that one could find this in the method of prime and
ultimate ratios. Berkeley therefore attacked this type of reasoning, as
applied by Newton in one of his leading demonstrations: that of finding
the fluxion of x n in De quadratura, in which Newton had sought to
avoid the infinitely small. 6 Here it will be recalled that Newton had
given x an increment o, had expanded (x + o) n by the binomial the-
orem, had subtracted x n to obtain the increment in x n , had divided
by o to find the ratio of the increments of x n and of x, and had then
* Cf. ibid., p. 30.
5 Ibid., pp. 22-23. 6 Ibid., Ill, 24-25.
226 The Period of Indecision
let o become evanescent, thus determining the ultimate ratio of the
increments (or of the fluxions). Berkeley averred that Newton here
disregarded the law of contradiction, assuming first that x has an
increment and then, in order to reach the result, allowing the incre-
ment to be zero, i. e., assuming that there was no increment. Berkeley
maintained that the supposition that the increments vanish destroys
the supposition that they were increments. Such an interpretation of
Newton's meaning, which of course results in the consideration of the
indeterminate ratio -, is not unjustified, inasmuch as Newton did
not sufficiently explain the terms "evanescent quantity" and "prime
and ultimate ratio," upon which the reasoning depends. The modern
interpretation in terms of limits, of course, considers the infinite
sequence formed by the ratios as the increment approaches zero, and
this has no last term, although it is defined as having a limit. Newton's
expression "ultimate ratio" is misleading, to say the least, and in all
events indicates a lack of appreciation of the subtle difficulties involved
in the concepts of infinity, continuity, and real number — difficulties
not resolved until the second half* of the nineteenth century.
Although Berkeley's arguments were directed chiefly against the
British method of fluxions, the method of differentials, as used on the
Continent by L'Hospital and others, also came in for criticism. 7 He
explained that in finding the tangent by means of differentials, one
first assumes increments; but these determine the secant, not the
tangent. One undoes this error, however, by neglecting higher differ-
entials, and thus "by virtue of a twofold mistake you arrive, though
not at science, yet at the truth." This interpretation of the validity of
the results of the calculus, as due to a compensation or errors, 8 we shall
find advanced again by Euler, Lagrange, and Carnot, proponents of
the differential method who wished to clarify its bases.
Berkeley's criticism of Newton's propositions was well taken from
a mathematical point of view, and his objection to Newton's infini-
tesimal conceptions as self-contradictory was quite pertinent. On the
other hand, his objections to some of Newton's quantities — such as
fluxions, nascent and evanescent augments, moments as increments
in statu nascenti, prime and ultimate ratios, infinitesimals, ultimate
1 1bid., Ill, 20, 29. « Ibid., Ill, 32.
The Period of Indecision 227
forms of evanescent triangles — on the grounds of their putative in-
comprehensibility or inconceivability, although just as much called
for, were really misdirected. Assuming that the symbols involved have
been clearly and logically defined (except for the primary undefined
elements of the subject), it is of no mathematical consequence whether
one can conceive of them in some manner corresponding to physical
perception. Thus Berkeley argued that the concept of velocity de-
pends upon space and time intervals and that it is consequently
impossible to conceive of an instantaneous velocity, i. e., of a velocity
in which these intervals are zero. 9 His argument is of course absolutely
valid as showing that instantaneous velocity has no physical reality,
but this is no reason why, if properly defined or taken as an undefined
notion, it should not be admitted as a mathematical abstraction. It is
interesting to notice that just as the arch-materialist Hobbes, being
unable to conceive of lines without thickness, denied them to geom-
etry, so also Berkeley, the extreme idealist, wished to exclude from
mathematics the "inconceivable" idea of instantaneous velocity. This
is in keeping with Berkeley's early sensationalism, which led him to
think of geometry as an applied science dealing with finite magni-
tudes which are composed of indivisible "minima sensibilia." 10
In line with this idea, Berkeley suggested that it would be better
in the calculus to consider increments than velocities. At any rate, he
warned, increments and velocities should not be confused as Newton
had upon occasion done. 11 He accepted Cavalieri's indivisibles, but
insisted that they were finite in number, saying that divisibility to
infinity is only a fiction, and that infinitely small magnitudes are in-
conceivable, for they imply the existence of extension without per-
ception by the mind through the senses. Berkeley was unable to
appreciate that mathematics was not concerned with a world of "real"
sense impressions. In much the same manner today some philosophers
criticize the mathematical conceptions of infinity and the continuum,
failing to realize that since mathematics deals with relations rather
than with physical existence, its criterion of truth is inner consistency
rather than plausibility in the light of sense perception or intuition.
Although those of Berkeley's arguments which are based upon the
9 Ibid., Ill, 19. 10 Johnston, The Development of Berkeley's Philosophy, pp. 82-86.
11 The Works of George Berkeley, III, 46-47.
228 The Period of Indecision
inconceivability of the notions involved lose their force in the light
of the modern view of the nature of mathematics, it is clear that there
was an obvious need for a logical clarification of many of the terms
Newton had used. Berkeley's animadversions, although those of a
nonmathematician, were successful in making this fact appreciated.
As a result, there appeared within the next seven years some thirty
pamphlets and articles which attempted to remedy the situation.
The first appeared in 1734, a pamphlet by James Jurin, Geometry No
Friend to Infidelity: or, A Defence of Sir Isaac Newton and the British
Mathematicians, in a Letter Addressed to the Author of the Analyst by
Philalethes Cantabrigiensis. This defense is weak in the extreme. Jurin
maintained categorically that fluxions are clear to those versed in
geometry. With respect to Berkeley's specific criticism on Newton's
determination of the moment of AB, Jurin gave two answers: the
moment ab in Berkeley's explanation he held to be as a pin's head
to the globe of the earth or of the sun or of the orb of the fixed stars;
the procedure of Newton, however, he defended by asserting that the
moment is the arithmetic mean of the increment and decrement! In
answer to Berkeley's objections to the determination of the fluxion of
x n as given by Newton in De quadratura, Jurin ingenuously said that
one is not to let the increment in this case be nothing, but to let it
"become evanescent" or "be upon the point of evanescence," affirming
that "there is a last proportion of evanescent increments." 12 Jurin's
response shows that he had no adequate appreciation either of Berk-
eley's arguments or of the nature of the limit concept.
Berkeley answered Jurin in 1735, in A Defence of Freethinking in
Mathematics, 13 and justly asserted that the latter was attempting to
defend what he did not understand. 14 In this work Berkeley again ap-
pealed to the divergence in Newton's views — as presented in De analysi,
the Principia, and De quadratura — to show a lack of clarity in the ideas
of moments, fluxions, and limits. Jurin's reply in the same year, in
The Minute Mathematician, was again evasively tautological. He
explained that "A nascent increment is an increment just beginning
to exist from nothing, or just beginning to be generated, but not yet
12 See Geometry No Friend to Infidelity, pp. 35, 52 ff . See also The Minute Mathematician,
p. 74.
13 Works, III, 61 ff. " Ibid., Ill, 78.
The Period of Indecision 229
arrived at any assignable magnitude how small soever." 15 By Newton's
ultimate ratio he understood literally "their ratio at the instant they
vanish." 16 Instead of explaining Newton's lemma on the moment of
a product in terms of limits, Jurin allowed himself to become involved
in the tangle Of infinitesimals and was forced to resort to the idea of
inassignables of Leibniz, saying that the magnitude of a moment is
nothing fixed or determinate, but is "a quantity perpetually fleeting
and altering till it vanishes into nothing; in short it is utterly unas-
signable." 17
Berkeley, whose Analyst "marks a turning-point in the history of
mathematical thought in Great Britain," 18 now dropped out of the
controversy, 19 but the unsatisfactory nature of Jurin's arguments
was pointed out by Benjamin Robins in A Discourse Concerning the
Nature and Certainty of Sir Isaac Newton's Methods of Fluxions and
of Prime and Ultimate Ratios, as well as in articles in current journals. 20
As indicated in the title of his book, Robins distinguished, not three
views in Newton's work, but two: that of fluxions and that of prime
and ultimate ratios. The former he considered to be the more rigor-
ous, saying that Newton used the latter only to facilitate demonstra-
tions. 21 He added that the method of fluxions is established without
recourse to the method of limits. 22 Robins admitted that Newton's
use of moments in the lemma of the second book of the Principia
was such as to allow interpretations resembling the language of
infinitesimals; but he said that Newton thought it sufficient to indi-
cate once for all that this can be made conformable to the method of
prime and ultimate ratios, as presented in the lemmas of the first
book. 23
Although Jurin had denied the possibility of infinitely small con-
stants, he had somewhat hazily espoused infinitely small variables, or
15 See The Minute Mathematician, p. 19. 16 Ibid., p. 30; cf. also p. 56.
"Ibid., p. 56.
18 Cajori, A History of the Conceptions of Limits and Fluxions, p. 89.
19 He had also answered the tract, A Vindication of Sir Isaac Newton's Principles of
Fluxions Against the Objections Contained in the Analyst, by J. Walton, but neither the
Vindication — a tautological paraphrase of Newton — nor Berkeley's answer contains any
significantly new views. See The Works of George Berkeley, III, 107.
20 See Robins, Mathematical Tracts, Vol. II. 21 Mathematical Tracts, II, 86.
» See Gibson, "Berkeley's Analyst," pp. 67-69.
» Cf. Mathematical Tracts, II, 68 ff .
230 The Period of Indecision
vanishing quantities. Robins was more emphatic in his disavowal of
infinitesimals of any kind and said that Newton's statements involving
moments are to be interpreted in terms of prime and ultimate ratios.
For example, whereas Jurin had said that Ab + Ba was equal to
Ab + Ba + ab when a and b banish, Robins said that Ab + Ba was
as much of the increment of AB as is necessary for expressing the ulti-
mate ratio. This indicates that Robins realized the more clearly that
the logical basis is to be found in the method of limits, although he
was not clear, inasmuch as the product AB involves two independent
variables, as to exactly how this was to be applied to the case in hand.
The limit conception of Robins represents a formulation of ideas
which Valerio and Tacquet had expressed somewhat vaguely a cen-
tury earlier — ideas to which Robins referred. 24 It indicates, as well,
the dependence of this notion, in his thought, upon geometric intui-
tion, for Robins spoke not only of the limits of "ratios of vanishing
quantities" but also of the limits of the "forms of changing figures,"
giving as an illustration the circle as the limit of the inscribed regular
polygon, as the number of sides is indefinitely increased. 25 This con-
fusion of the arithmetical with the geometrical had been responsible
for much of the vagueness in the work of Newton and of Leibniz,
and was to persist during the following century. Nevertheless, Robins
realized more clearly than did Jurin the nature of the limit concept.
He recognized that the phrase "the ultimate ratio of vanishing quan-
tities" was a figurative expression, referring, not to a last ratio, but
to a "fixed quantity which some varying quantity, by a continual
augmentation or diminution shall perpetually approach, . . . pro-
vided the varying quantity can be made in its approach to the other
to differ from it by less than by any quantity how minute soever,
that can be assigned," 26 . . . "though it can never be made abso-
lutely equal to it." 27
Robins realized, as Jurin did not, that the varying quantity need
not be considered as finally reaching the fixed quantity as its last
value, although this latter "is considered as the quantity to which the
varying quantity will at last or ultimately become equal." 28 In the
controversy between Robins and Jurin, the question as to whether a
u Ibid., II, 58. «IWd.,II,54.
26 Mathematical Tracts, II, 49. a Ibid., II, 54. » Ibid.
The Period of Indecision 231
variable was to be considered as necessarily reaching its limit played
a large part. Robins upheld the negative side; Jurin insisted that there
are variables which reach their limits and vigorously accused his
opponent of misinterpreting Newton's true meaning. It is difficult to
judge from Newton's words exactly what he meant. The phrase
"ultimate ratio" certainly favors Jurin's interpretation, but to avoid
the logical difficulties inherent in questions of infinitesimals and the
meaning of -, it was necessary at the time to accept the more logical
view of Robins that the variable need not attain its limit.
Robins has recently been criticized 29 for defining his limit in such a
way that the variable never reaches its limit, on the grounds that
although this has certain pedagogical advantages, it involves a less
general conception of limit than Jurin's, and that under this con-
ception of a limit Achilles could not overtake the tortoise. This
criticism has been followed by the assertion that one can assume a
time rate of doubling of the number of sides of a polygon inscribed in
a circle, such that the circumference (the limit) is reached by the
polygon (the variable). Such an argument is entirely beside the
point. Besides confusing the numerical concept of limit with a geo-
metrical representation, it merely substitutes an infinite time series
for Zeno's distance series. This it does under the assumption, ap-
parently, that this is intuitively more impelling, because of our vague
idea of the relentless flow of time, which in the Achilles is subordinated
to the static idea of distance. The question as to whether the variable
S n reaches the limit S is furthermore entirely irrelevant and ambiguous,
unless we know what we mean by reaching a value and how the terms
"limit" and "number" are defined independently of the idea of
reaching. Definitions of number, as given by several later mathe-
maticians, make the limit of an infinite sequence identical with the
sequence itself. Under this view, the question as to whether the
variable reaches its limit is without logical meaning. Thus the infinite
sequence .9, .99, .999 . . . , is the number one, and the question,
"Does it ever reach one?" is an attempt to give a metaphysical argu-
ment which shall satisfy intuition. Robins could hardly have had
29 See Cajori, A History of the Conceptions of Limits and Fluxions, Chap. IV.
232 The Period of Indecision
such a sophisticated view of the matter, but he apparently realized,
as Jurin does not appear to have done, 30 that any attempt to let a
variable "reach" a limit would involve one in the discussion as to the
nature of -. Thus he is hardly to be criticized for his restriction.
We shall find that the question as to whether a variable reaches its
limit or not has no significance, in the light of modern definitions.
At that time, however, it was important as an indication that mathe-
maticians still felt that the calculus must be interpreted in terms of
what was intuitively reasonable, rather than of that which was log-
ically consistent. This is apparently the reason why Robins considered
the method of fluxions more satisfactory than that of prime and
ultimate ratios. Everyone assumed he had a clear idea of instantaneous
motion, although logically this is defined, as Robins failed to realize,
in terms precisely equivalent to those needed to make the idea of prime
and ultimate ratio rigorous.
That the work of Robins was not fully appreciated in England is
seen in the fact that although Berkeley's Analyst was frequently
discussed in the flood of texts on fluxions appearing in 1736 and
1737, the Jurin-Robins controversy was not referred to. 31 Mathema-
ticians were still not satisfied with the method of limits and, although
the discussion on infinitesimals may have discouraged the use of the
idea of moments, it did not banish them. In 1771 the article on fluxions
in the Encyclopedia Britannica read: "The fluxion of any magnitude
at any given point is the increment that it would receive in any given
time, supposing it to increase uniformly from that point; and as the
measure will be the same, whatever the time be, we are at liberty to
suppose it less than any assigned time." 32 The old confusion between
fluxions and moments had not yet ended, and Robins' clarification of
Newton's method of prime and ultimate ratios was not sufficient to
establish this as the basis until after continental influences, in the
early nineteenth century, had brought about a turning point in British
mathematics.
In the meantime, however, numerous attempts, some noteworthy
30 See Jurin, "Considerations upon Some Passages of a Dissertation Concerning the
Doctrine of Fluxions Published by Mr. Robins," pp. 68 ff.
11 Cajori, A History of the Conceptions of Limits and Fluxions, p. 179.
M /Wtf.,p. 240.
The Period of Indecision 233
and others insignificant, were made to find new and more satisfactory-
forms and arguments in which to present Newton's method. By far
the ablest and most famous of these was made by Colin Maclaurin.
In his Treatise of Fluxions, in 1742, he aimed, not to alter the con-
ceptions involved in Newton's fluxions, but to demonstrate the
validity of his method by the rigorous procedures of the ancients 33 —
to deduce the new analysis from a few "unexceptional principles." 34
Maclaurin professed in the preface of this work that the Analyst
controversy had given occasion to his treatise. Therefore he proceeded
with extreme circumspection, omitting the notation of fluxions until
toward the end of the long two-volume treatise. Like Robins, he
banished the infinitely small as inconceivable and as "being too bold
a Postulatum for such a Science as Geometry." 35 He did not, however,
see any objection to introducing into geometry the idea of an instan-
taneous velocity, for he felt that there can be no difficulty in con-
ceiving velocity wherever there is motion. 36 In fact, the mathematical
sciences, Maclaurin said, included velocity and motion, as well as the
properties of figures. 37 Time, he conceived with Barrow, to flow in a
uniform course that serves to measure the changes of all things. 38
Barrow, however, had defined velocity as the power by which a
certain space may be described in a certain time, 39 somewhat as
Aristotle had considered motion a manifestation of a potentiality.
Maclaurin, on the other hand, tried to define instantaneous velocity
in a manner recalling the attempt of Oresme: "The velocity of a
variable motion at any given term of time is not to be measured by
the space that is actually described after that term in a given time,
but by the space that would have been described if the motion had
continued uniformly from that term." 40 Maclaurin realized that if an
instantaneous speed is "susceptible of measuring, it is only in this
sense." 41 He recognized, that is, that science deals only with actual
intervals; but he failed to see that as a mathematical notion instan-
taneous velocity could be defined by extrapolating beyond sense im-
pressions, through the limit of an average rate of change as the inter-
vals approach zero.
Although Maclaurin considered his explanation of fluxions as a
83 A Treatise of Fluxions, I, 51 ff.
34 Ibid., I, Preface.
3S Ibid.,I, iv.
*<Ibid., I, iii.
« Ibid., I, 51.
3 * Ibid., I, S3.
*/fttf.,I,54.
40 Ibid., I, 55.
"Ibid.
234 The Period of Indecision
criticism of the method of differentials, his interpretation of fluxions,
in terms of the intervals which would be generated if the motion were
to continue uniformly, left the way open to an explanation of Newton's
fluxionary procedures in terms of finite differences and limits, by which
the differential calculus of Leibniz was also to be explained. Brook
Taylor had recognized the importance of this type of exposition and
had some years previously composed a book on the subject — his
Methodus incrementorum directa et inversa.
Whereas Robins and Maclaurin emphasized in Newton's work the
interpretations in terms of fluxions, Taylor said that Newton had
founded his method on prime and ultimate ratios. 42 Newton had
recognized that the limit of the ratio of the moments was the same
as the ratio of the corresponding fluxions or velocities. Whereas Robins
and Maclaurin had assumed that everyone had a clear idea of instan-
taneous velocity, Taylor felt that it was easier to conceive of moments
and to obtain the ratios of fluxions from these. 43 In this respect, how-
ever, his work at times approached closely to an unclear manipulation
of zeros, resembling the later procedures of Euler. Taylor said that the
relation of fluxions was to be obtained from that involving finite
differences. In this respect his view resembled that of modern mathe-
maticians, although Leibniz said at the time that it was "putting the
cart before the horse." 44 However, Taylor, like Leibniz, was not clear
with respect to the transition from finite differences to fluxions, for
he held that to bring this about one simply wrote zero for the "nascent
increments." Ultimate ratios, he thought, are those in which the
quantities are already evanescent and are made zero, 45 an attitude
which was to appear on the Continent in the work of Euler.
The view of Taylor on instantaneous velocities was shared by
Thomas Simpson, author of a popular textbook on the method of
fluxions which appeared first in 1737 and in an enlarged edition in
1750. Simpson felt that by taking fluxions as mere velocities, the
imagination is confined to a point, and, without proper care, insen-
sibly involved in metaphysical difficulties. 46 Maclaurin recognized this
difficulty also and said that fluxions were measured by the quantities
42 Taylor, Methodus incrementorum directa et inversa, Preface. **Ibid.
44 Malhematische Schriften, III (Part 2), 963.
45 Methodus incrementorum directa et inversa, Preface, and p. 3.
46 The Doctrine and Application of Fluxions, pp. xxi-xxii.
The Period of Indecision 235
they would generate, if they were to continue uniformly. Simpson
went beyond Maclaurin and identified fluxions of accelerated quan-
tities with the increments which would be generated in a given portion
of time if the "generating celerity" were to continue uniformly. 47
Simpson therefore followed Taylor in reverting to Newton's use of
moments. He employed v in place of Newton's o, and as Newton had
in his De analyst omitted terms containing o, so Simpson dropped
out powers of v when the points coincided, because these powers he
felt were due to the acceleration of the motion, 48 whereas, according
to his definition, the generating celerity was to be regarded as con-
tinuing uniformly.
The views of Taylor and Simpson are similar to those which Jurin
expressed in opposing Robins and serve to indicate the continued
recurrence and widespread use of infinitesimal quantities in England
during the century following the time of Newton, and the confusion
of these with fluxions. There was also another element in British
thought which operated against an early clarification of the bases of
the calculus. The method of differences of Taylor failed to exert a
decisive influence not only because of the novelty of his notation and
his lack of clarity of expression, but also because it was essentially
arithmetical and so involved a degree of abstraction which seems to
have been unwelcome in British mathematics at the time. The basis
of the method of fluxions remained essentially geometrical in its
conception of quantity, in spite of Newton's work in infinite series;
and the tenacity with which English mathematicians clung to the
idea of velocity was probably due as much to a desire for an intuitively
satisfying conceptual background as to loyalty to their great prede-
cessor. This is shown particularly in the treatise of Maclaurin, which
represents the high point in the rigorous interpretation of the calculus
in terms of geometrical and mechanical notions. This work, however,
was as little read as it was widely praised, and it consequently had
probably no more influence than had Robins' treatises, which it
resembles in its general ideas.
Much of the Analyst controversy and the confusion in the inter-
pretation of the limit concept was due to the lack of a clear distinction
between questions of geometry and those of arithmetic, and to the
47 Ibid., p. 1; cf. also p. xxii. a Ibid., pp. 3-4.
236 The Period of Indecision
absence of the formal idea of a function. These weaknesses in the
English view are brought out particularly in Newton's determination
of the moment of a product, which was the object of some of Berkeley's
most pertinent strictures. Newton had indicated vaguely that this was
to be made rigorous by the application of the limit method, which is
essentially arithmetical, but he had interpreted the product geo-
metrically as the area of a rectangle. As a consequence, there was no
indication as to whether the product was to be considered a function
of one or of two variables. Time was, in a sense, made to take the
place of an independent variable, but this was merely to aid in the
conceptual representation and not to reduce the problem to one
expressible in terms of a function of a single independent variable,
as is necessary for differentiation.
We shall find that on the Continent there was a growing tendency
to link the calculus with the formal concept of a function, instead of
with the intuitional conceptions of geometry. The ideas developed as
a result of this trend were not significantly influential in changing
British views until the beginning of the following century, but mean-
while there were in England occasional abortive attempts to sub-
stitute arithmetical devices and ideas for those of geometry and
dynamics. 49 Here and there the idea was expressed that the operations
of the new analysis should proceed along the line afforded by the ordi-
nary methods of arithmetic and algebra, and that the introduction of
the doctrine of motion was unwarranted and unnecessary. However,
the efforts to establish an algebraic calculus were characterized by
lack of rigor in exposition.
One of the best known and least objectionable of the efforts to base
the calculus only upon principles received in algebra and geometry,
"without the aid of any foreign ones relating to an imaginary motion
or incomprehensible infinitesimals," was made in 1758 by John Landen
in what he called The Residual Analysis. Instead of computing the
quotient of fluxions, or of differentials, Landen calculated "the value
of the quotient of one residual divided by another." 50 By a residual
was to be understood an expression of the form x — x, or x n — x n .
Landen's method thus was based upon an uncritical manipulation of
49 See Cajori, A History of the Conceptions of Limits and Fluxions, Chap. IX.
60 The Residual Analysis, p. v.
The Period of Indecision 237
indeterminate forms. Given a function F(x), Landen found "we shall
frequently have occasion to assign the quotient of F — F divided by
m
x — x. bl For example, the fluxion, or residual quotient, of x n he found
by writing
v fv\ 2 IV
x — v - - 1
= x n
X — V
1+^ + 1, !■+•-• +
x \xl \x/
,' + (:)H)* + (f-
and then taking v = x. b2
The method of Landen has been characterized as making use of
the limit of D'Alembert supposed to be attained instead of a terminus
which can be approached as closely as desired. 53 Such a judgment is
most charitable. If Landen possessed the limit concept, he certainly
hid it most effectively under a misleading notation and terminology
at a time when there was need for a clear and open recognition of its
fundamental importance. Although we shall find that D'Alembert
was at the time urging upon continental mathematicians that the
logical basis of the differential calculus was to be found in the notion
of limits, there was in England no strong leader to propagate this
doctrine, unhampered by geometrical and mechanical superfluities.
When it finally imposed itself upon British mathematics, it was as
the result of European developments, to which we must now turn.
While the English mathematicians were so greatly occupied with
arguments as to the validity of the views involved in the method of
fluxions, the differential calculus was rapidly gaining in popularity on
the Continent. The algorithmic essentials of the differential calculus
of Leibniz had appeared in the Acta eruditorum for 1684, and those
of the "calculus summatorius," or integral calculus, followed in 1686.
Inasmuch as Leibniz, unlike Newton, corresponded extensively with
numerous mathematicians on the subject of the new analysis, seeking
for the most suitable forms of notation and presentation, there grew
up a group of enthusiastic admirers of the subject who were soon
able to make contributions of their own.
61 Ibid., p. 5.
62 Ibid., pp. 5-6; see also A Discourse Concerning the Residual Analysis, pp. 5, 41.
63 See Cajori, A History of the Conceptions of Limits and Fluxions, pp. 238-39.
238 The Period of Indecision
In Switzerland, for example, there were John and James Bernoulli,
the former of whom in 1691-92 wrote a little treatise on the differ-
ential calculus, although this was not published until 1924. 54 The
first published textbook on the subject appeared in 1696. This was
by a French disciple of Leibniz, the Marquis de l'Hospital, and the
title is characteristic of his approach — Analyse des infiniments petits
pour V intelligence des lignes courbes. This work is based, at least in
part, on the earlier work of John Bernoulli. 55 Although l'Hospital did
not in this book discuss the nature of the basic concepts of the cal-
culus, he played a significant r61e in the popularization of the new
subject, both through the fact that his text appeared in numerous
editions 56 and through the influence he exerted in the Journal des
savants} 7 Through this journal and the Acta eruditorum was created
an atmosphere of enthusiasm for the differential calculus which led
to a disregard on the Continent of the method of fluxions. It is inter-
esting to notice that this parallels the comparative lack of regard for
Newtonian science, in favor of that of Descartes, until the former
was popularized in France by Voltaire. In Germany the differential
calculus was popularized in the philosophical works of Christian
Wolff, as well as by the mathematical work of the Bernoulli brothers.
In Italy an enthusiastic interest in the differential calculus was mani-
fested by Guido Grandi, another of the correspondents of Leibniz and
author of a number of works on the calculus during the early eight-
eenth century.
In spite of the popularity which the calculus of Leibniz enjoyed,
there was a total lack of clarity and agreement as to the basis of the
analysis. Voltaire called the calculus "the Art of numbring and meas-
uring exactly a Thing whose Existence cannot be conceived." 88 The
indecision which Leibniz had displayed was shared also by his fol-
lowers. John Bernoulli's little volume on the differential calculus
begins with the paradoxical postulate that a quantity which is dimin-
ished or increased by an infinitely small quantity is neither diminished
64 See Die Differenzialrechnung von Johann Bernoulli.
M See Enestrom, "Sur la part de Jean Bernoulli dans la publication de 1' Analyse des
infiniment petits"; cf. Rebel, Der Briefwechsel zwischen Johann (I.) Bernoulli und dent
Marquis de l'Hospital, pp. 9 ff.
68 Paris, 1715, 1720, and 1781; Avignon 1768.
67 Sergescu, "Les Mathematiques dans le 'Journal des Savants,' 1665-1701."
68 See Letters Concerning the English Nation, p. 152.
The Period of Indecision 239
nor increased. 69 In other words, Bernoulli made fundamental the
omission of differentials of higher orders, 60 rather than the limit con-
cept. Similarly, in the integral calculus the figure bounded by an
infinitely small piece of a curve, the ordinates of its end points, and
the corresponding difference in the abscissas he considered as a par-
allelogram. 61 Although he thought of a surface as the sum of such
differentials of area, he did not define the integral as such a sum, as
had Leibniz, but rather as the inverse of the differential, with the
addition of a suitably chosen constant 62 — a definition which persisted
throughout the next century.
We have seen that Leibniz generally considered his differentials as
only indefinitely or imcomparably small, but John Bernoulli boldly
asserted in a letter to Leibniz in 1698 that inasmuch as the number
of terms in nature is infinite, the infinitesimal exists ipso facto™
This assertion he attempted to clarify in a manner recalling the exposi-
tion of Pascal in which the latter had pointed out that, through the
reciprocal relationship, the existence of the indefinitely small was
implied by that of the indefinitely great. Bernoulli sought to apply
this type of argument to the case of actual infinitesimals: Let the
infinite series i, i, §, . . . be given. Then if there are ten terms, one-
tenth exists; if there are a hundred, then a hundredth exists; ... if
the number of terms is infinite, as is here supposed, the infinitesimal
exists. 64 Leibniz, in answering, wisely cautioned him that arguments
concerning the finite need not hold for the infinite, and that further-
more the infinite and the infinitesimal may be imaginary, even though
they determine real relationships. 65 John Bernoulli, nevertheless, dis-
regarded this caveat and persisted in views with respect to the infinitely
large and the infinitely small which call to mind the earlier work of
Wallis.
Of the two famous Bernoulli brothers, John had the greater origi-
nality and imagination, but James was superior in critical power. 66
69 John Bernoulli, Die Dijferenlialrechnung, p. 11.
60 Cf. Leibniz, Mathematische Schriflen, III (Part 1), 366.
61 John Bernoulli, Die Dijferentialrechnung, p. 11.
62 John Bernoulli, Die erste Iniegralrechnung, pp. 3, 8, 11-12.
63 Leibniz, Mathematische Schrijten, III (Part 2), 555.
64 Ibid., pp. 563 ff.; see also Leibniz and John Bernoulli, Comnterciutn philosophicum et
malhematicum, I, 400-31.
68 Commerciiim philosophicum et mathematicum, I, 370.
M Mach, The Science of Mechanics, pp. 427-28.
240 The Period of Indecision
This fact is well illustrated by their respective attitudes toward the
calculus. Whereas John expressed the positive attitude of Leibniz with
reference to infinitesimals, his brother James put forth Leibniz's more
cautious view. He found the use of the infinite not sufficiently con-
vincing and too far from the opinion of the ancients. 67 He held that the
infinitely small was not to be thought of as a determined quantity,
but as a fiction of the spirit — "a perpetual fluxion toward nothing."
2yy -f- dy 2
As a consequence, "the ratio is always variable and does
4yy — dy 2
not become fixed unless dy is perfectly zero." 68 This view of the
differential as a variable would associate the calculus with the method
of limits, but James Bernoulli was unable to express this notion
clearly because he failed, as had Leibniz, to distinguish between
independent and dependent variables. In other words, the function
concept had not yet become primary.
Although James Bernoulli attempted to avoid the pseudo-infinites-
imal and maintained that a quantity smaller than any given magnitude
is zero, 69 he vacillated in his attitude. On occasion he asserted that the
Euclidean axiom "If equals are taken away from equals, the results
are equal," need not be absolutely true when incomparably small
quantities are involved. 70 For this reason he gave warning that in the
calculus of the infinitely small, one must proceed with caution to
avoid paralogisms. 71
Wolff, a follower of Leibniz in mathematics and philosophy whose
works enjoyed a wide distribution, adopted a modification of the views
of James Bernoulli which the writings of Leibniz had suggested. He
thought of the infinitely large and the infinitely small as impossi-
bilities or as convenient geometrical fictions, useful for discovery,
which result from a figurative manner of speaking. 72 By calling some-
thing infinitely large, he said, one simply means that it can exceed
any number. Similarly, the infinitely small is not really a quantity in
the strict sense of the word, but rather some sort of imaginary sym-
bolism as Leibniz had sometimes held. 73
If some mathematicians followed Wolff in denying the reality of the
67 Leibniz, Mathematische Schriften, V, 350.
6? Leibniz, Mathematische Schriften, III (Part 1), 52-56.
69 Opera, I, 379. 70 Opera, II, 765. n Ibid.
72 Philosophia prima, sive ontologia, pp. 597-602. n Ibid.
The Period of Indecision 241
infinitely small, others took the opposing view. In Italy Guido Grandi
upheld the existence of absolutely infinite and infinitesimal magni-
tudes of various orders. Those of first order he defined as quantities
which bear to any finite magnitude of the same kind a ratio respec-
tively greater and less than any assignable number. 74 Those of higher
order were similarly defined in terms of those of lower order. 75 Quan-
tities differing by less than any assignable magnitude he considered
equal, because this, he felt, represented only a short way of saying
what Euclid and Archimedes had meant in their work with in and
circumscribed figures. 76 As an example of the addition of differentials
to give a finite magnitude, Grandi referred to the paradoxical result
l-l + l-l+... = + + ... = i. This, he suggested to
Leibniz, could be compared with the mysteries of the Christian
religion and with the creation of the world by which an absolutely
infinite force created something out of absolutely nothing. 77
In France, after a period of hesitation, views more bold even than
those expressed by Grandi were to appear. L'Hospital, the first student
of John Bernoulli, 78 had presented the views of his teacher in his
Analyse des infiniments petits; and a little later Pierre Varignon,
Bernoulli's "best friend in France," 79 worked for the new analysis.
However, as Nieuwentijdt had opposed the calculus of Leibniz and
Berkeley that of Newton, so there arose in 1700, in the French
Academie des Sciences, a lively dispute as to the validity of the
infinitesimal methods. 80 In this discussion Rolle maintained that the
new methods led to paralogisms, while John Bernoulli maintained in
an argumentum ad hominem that Rolle did not understand the cal-
culus. 81 Varignon attempted to clarify the situation by showing indi-
rectly that the infinitesimal methods could be reconciled with the
geometry of Euclid. 82
In 1727, however, the French author Bernard de Fontenelle, a
friend of Varignon, could boast that there were no longer two parties
74 De infiniiis infinitorum et infinite parvorum ordinibus, pp. 22-23.
75 Ibid., pp. 26 ff. 7 « Ibid., p. 39.
77 Cf. Leibniz, Mathematisehe Schriften, IV, 215-17; Reiff, Geschichte der unendlichen
Reihen, p. 66.
78 Fedel, Der Briefwechsel Johann (i) Bernoulli-Pierre Varignon, p. 3. n Ibid., p. 2.
80 See a note in Histoire de I' Academie des Sciences, 1701, pp. 87-89.
81 Leibniz, Mathematisehe Schriften, III (Part 2), 641-^2.
w Fedel, Der Briefwechsel Johann (i) Bernoulli-Pierre Varignon, p. 25.
242 The Period of Indecision
in the Academic He evidenced this fact by publishing his Siemens de
la geomeirie de I'infini, in which there is no semblance of doubt ex-
pressed on the subject. The work of Fontenelle displays an absolute
dogmatism with respect to the infinite. Recognizing that geometry is
entirely intellectual and independent of the actual description and
existence of figures, 83 Fontenelle did not discuss the subject from the
point of view of science or metaphysics, as had Aristotle and Leibniz.
He objected to regarding the infinite as a mystery and protested
that Cavalieri was too modest in his treatment. 84 Confidently follow-
ing Wallis, Fontenelle wrote oo as the last term of the infinite
sequence 0, 1, 2, 3, . . ., although he realized that the manner in
which the series goes from the finite to the infinite is inconceivable. 85
On the basis of this definition of the infinite, Fontenelle went on to
include in his calculation not only integral powers of °° , but even
fractional and infinite powers as well, using such symbols as °° * and
oo °° 3 , and writing such equalities as oo • °° * ~ x = °° °°. 86 As Wallis
had written the infinitely small as J_, so Fontenelle derived his orders
oo
of infinitesimals as the reciprocals of the powers of infinity. The
differentials dy and dx he held to be magnitudes of the order J_,
oo
although he defined these in terms of the characteristic triangle of
Leibniz. 87
It is interesting to note that we have found three men — Wallis,
John Bernoulli, and Fontenelle — who tried in an arithmetic manner
to derive the infinitely small as the reciprocal of the infinitely large.
(Pascal had in this manner sought only to relate the indefinitely large
to the indefinitely small.) Such attempts lacked all semblance of
mathematical rigor because of the lack at that time of satisfactory
definitions of cither the infinite or the infinitesimal. These efforts were
furthermore counter to the general tendency of the time, which found
the basis of mathematics in geometrical conceptions. Arithmetic had
not become sufficiently abstract and symbolic to free itself of spatial
interpretations, for number was still interpreted metrically as a ratio
of geometrical magnitudes. Descartes had affirmed the identity of
numerical and geometrical calculations. Newton had said that a num-
83 Elemms de la geometrie de Vinfini, Preface. M Ibid.
85 Ibid., pp. 30-31. » 6 Ibid., pp. 40 ff. * Ibid., p. 311.
The Period of Indecision 243
ber was a ratio of quantities, and Wolff wrote that a number was
anything which referred to unity in the same way as one straight line
to another. 88 The methods of fluxions and differentials were, as a
result, naturally considered convenient processes for solving geomet-
rical problems. Although the results were usually expressed in algebraic
terminology, the bases were sought in the geometry of the ancients,
rather than in arithmetical conceptions. With the greatest mathe-
matician of the first half of the eighteenth century, however, a change
of view — anticipated to some extent by Wallis, Bernoulli, and Fon-
tenelle — entered the new analysis.
Leonhard Euler wrote a prodigious number of books and articles
"promoting the new analysis, organizing it, and putting it on a formal-
istic basis. Most of his predecessors had considered the differential
calculus as bound up with geometry, but Euler made the subject a
formal theory of functions which had no need to revert to diagrams
or geometrical conceptions. 89 Leibniz had used the word function
somewhat in our sense, and had boasted that his infinitesimal method
was not limited to algebraic functions, as was that of Descartes, but
was applicable to logarithms and exponentials as well. Nevertheless,
Euler was the first mathematician to give prominence to the function
concept and to make a systematic study and classification of all the
elementary functions, together with their differentials and integrals.
The word function, however, meant for Euler not so much any
quantity conceived as depending on variables, as an analytic ex-
pression in constants and variables which could be represented by
simple symbols. 90 Functionality was a matter of formal representation,
rather than conceptual recognition of a relationship. The almost
automatic development of the calculus during the eighteenth century
was largely the result of this formajistic view, to which the notation
of Leibniz was so remarkably well adapted. However, the greater the
success achieved by the differential calculus, the less constrained
Euler felt to justify his procedures. His views on the bases of the
subject were elementary in the extreme, resembling somewhat those
^Elemenla matheseos universalis, I, p. 21; cf. also Pringsheim, "Nombres irrationnels
et notion de limite," p. 144 n.
89 See the introduction to his Institutiones calculi differentiate . Opera omnia, Vol. X.
90 See Brill and Noether, "Die Entwickelung der Theorie der algebraischen Funktionen
in alterer und neuerer Zeit."
244 The Period of Indecision
of Wallis, Taylor, John Bernoulli, and Fontenelle. He felt that the
notions of the infinitely great and the infinitely small did not hide so
great a mystery as was commonly thought. An infinitely small or
evanescent quantity he held to be simply one which will be zero. 91
This view might well have served as the basis for an interpretation
in terms of limits, in which the differentials are simply variables ap-
proaching zero as a limit. Euler, however, did not proceed in this
manner. Throughout the development of the calculus, the pandemic
infinitesimal had, at various stages, been taken as a constant quantity
less than any assignable magnitude. Euler emphatically rejected any
such notion of mathematical atomism or monadology, inveighing
against this as a "wretched abuse of the principle of sufficient reason." 92
He asserted, as had James Bernoulli, that a number less than any
given quantity must of necessity be zero. 93 The differentials dx and
dy were therefore simply zero. 94 Although he admitted the existence
of an infinite number of infinitesimals, as found in the differentials of
higher orders, these he held were all zero. 95 Leibniz had at one point
suggested that the differentials could be regarded as qualitative zeros,
which nevertheless retained by the law of continuity the character
of the relationships of the finite quantities from which they were
derived. Euler, in conformity with his formalistic view, held less
philosophically that the zeros represented by differentials were to be
distinguished through the recognized fact that the ratio - could, in
n
a sense, represent any ratio of finite numbers, ~. 96 Thus for Euler the
calculus was simply the determination of the ratio of evanescent
increments — a heuristic procedure for finding the value of the ex-
pression ~. 97
91 Opera omnia, X, 69.
n Letters to a German Princess, II, 61 ; cf . also Opera omnia, X, 67.
* Opera omnia, X, 69-70. " Ibid., pp. 70-72.
96 Ibid.; cf . also XI, 5. »« Ibid., X, 70.
97 Cohen (Das Princip der Infinitesimalmethode und seine Geschichte, p. 96) displays a
lack of understanding of the limit concept in saying, in connection with Euler's use of
zeros: "Es ist offenbar, dass er hierein nur der Grenzmethode folgt, dieselbe aber iiber-
bietet: indem er das Inkrement selbst, nicht den Vorgang, als verschwindend (incrementum
evanescens) annimmt, als Null."
The Period of Indecision 245
In justifying the omission in the differential calculus of infinitesimals
of higher order, Euler's argument again lacked clarity. He held that,
in the expression dx =*= dx 2 , the infinitely small quantity dx 2 vanishes
before dx does, so that for dx = the ratio of dx ± dx 2 to dx will be
one of equality. 98 At one point he suggested that in omitting terms
involving differentials, one made allowance for certain errors," as
Berkeley had maintained; and on occasion he employed difference
calculus as a practical substitute for the differential calculus. 100
Euler, however, did not justify the transition from one to the other.
Leibniz had explained the substitution of inassignables for assignables
as validated by the law of continuity, but Euler followed Taylor in
simply substituting zero for the increments. For example, in deter-
mining the differential of x 2 he allowed x to become x + w . The ratio
of the increments in x and x 2 was then 1 : 2x + &>. This is always
different from the ratio 1 : 2x unless w vanishes. Euler therefore sub-
dx 2 2x
stituted for w and thus obtained the evanescent ratio — = — in
dx 1
much the same manner as Taylor had found the fluxion of a: 2 . 101
With respect to the infinite, Euler adopted the views of Wallis and
Fontenelle. Inasmuch as the sum of the series 1 + 2 + 3 + . . . can
be made greater than any finite quantity, it must be infinite and can
be represented by the symbol °° , 102 At another point he suggested
that oo W as a sort of limit between the positive and the negative
numbers, in this respect resembling the number 0. In a similar manner
a . .
he held that the relationship - = °° was to be interpreted as meaning
that nothing times infinity can result in a finite magnitude. 103 Further-
more, as — will be infinite because dx = 0, so - — will be infinite of
dx dx 2
second order; and, more generally, corresponding to the orders of the
differentials, there is an infinite number of grades of infinity. 104 If x
is infinite, he held that between 1 and x 1 ' 000 there are 1,000 grades of
98 Opera omnia, X, 71.
99 Cf. Weissenborn, Die Principien der hbhoren Analysis, p. 158.
100 Cf. Bohlmann, "tjbersicht iiber die wichtigsten Lehrbiicher der infinitesimalrechnung
von Euler bis auf die heutige Zeit."
101 Opera omnia, X, 7.
1W Ibid., p. 69. 1W Ibid., p. 73. m Ibid., p. 75.
246 The Period of Indecision
infinity. 105 The lack of care with which Euler handled the infinite is
evidenced also in his use of divergent series. As Leibniz had suggested
1
that 1-1 + 1-1 + ... = |, so Euler held that from
(1 + 1)'
= j one could conclude that 1 — 2 + 3— 4 + 5 — ... = £
Under a somewhat different point of view Euler added that 1 — 3 +
5 — 7 + . . . = 0. Numerous similar examples of divergent series are
to be found in his work. 106
Inasmuch as Euler restricted himself to well-behaved functions, he
did not become involved in those subtle difficulties connected with
the notions of infinity and continuity which were later to make such
a naive position untenable. Although his views on the fundamental
principles of the calculus lacked all semblance of the precision and
rigor which entered mathematics in the following century, the formal-
ists tendency which his work inaugurated was to free the new analysis
from all geometrical fetters. 107 It also made more acceptable the arith-
metic interpretation which was later to clarify the calculus through
the limit concept which Euler himself neglected.
While Euler, under the influence of Leibniz and the Bernoullis, was
working so successfully on the assumption that differentials were
zeros, his contemporary, Jean le Rond D'Alembert, was promulgating
the doctrine which was to be elaborated ultimately into that accepted
at the present time. Although the Newtonian-Leibnizian priority con-
troversy had estranged British and Continental mathematicians,
neither group was completely unaware of the views of the other.
Thus Robins had, in 1 739, criticized Euler for his crude conception of
the infinitesimal, saying that the error of his ways was due to following
"that inelegant computist" (John Bernoulli) who instructed him. 108
In like manner the substance of the Berkeley- Jurin-Robins con-
troversy was known to mathematicians on the Continent. Buffon, in
the historical introduction to his translation of Newton's Method of
Fluxions, criticized Berkeley and Robins for taking exception to some
of Newton's arguments and warmly espoused Jurin's weak and prolix
defense as "solid, brilliant, admirable." 109 Although he felt that
105 Opera omnia, XV, 298. 10S See Opera omnia, XIV, 585 ff.
m Cf . Merz, A History of European Thought in the Nineteenth Century, I, 103.
108 Cajori, A History of the Conceptions of Limits and Fluxions, pp. 139-40.
109 La Methode des fluxions et des suites infinies, pp. xxvii-xxix.
The Period of Indecision 247
Robins had criticized Euler and Bernoulli unfairly, 110 Buffon himself
opposed Euler's views on the infinite and the infinitesimal. He held
that the sequence 1, 2, 3, . . . had no last term and that the infinitely
large and the infinitely small were only "privations." 111 Buffon, more
interested in the natural than the mathematical sciences, did not elab-
orate this view; but the doctrine that the terms infinitely large and
infinitely small signified only indefinitely large and indefinitely small
was expounded more fully by the mathematician D'Alembert, who
made this the basis of his theory of limits.
D'Alembert was probably led by two earlier works to regard the
method of limits as fundamental in the calculus. These books, men-
tioned in his article on the differentiel in the famous Encyclopedic,
were Newton's De quadratura curoarum of 1704 and De la Chapelle's
Institutions de geometric of 1746, a popular text which connected him
with the adumbrations of the limit idea by Stevin, Gregory of St.
Vincent, and others. 112
D'Alembert interpreted Newton's phrase "prime and ultimate
ratio" not literally, as a first or last ratio of two quantities just spring-
ing into being, but as a limit. One quantity he called the limit of
another if the second can approach the first nearer than by any given
quantity, or so that the difference between them is absolutely inas-
signable. Properly speaking, however, he felt that the varying quan-
tity never coincides with, or is equal to, its limit. 113 D'Alembert thus
agreed essentially with Robins' interpretation of Newton's meaning.
He applied the same idea to the differential calculus. It will be recalled
that Leibniz had believed that one could think of differentials as
inassignable quantities, which by the law of continuity express the
ultimate relationship between quantities deprived of magnitude.
However, just as D'Alembert had denied the validity of the concep-
tion of an ultimate ratio, so also he rejected that of an ultimate
relationship which can be determined by differentials. He said that
the basis of the differential calculus, like that of the method of fluxions,
was to be found in the idea of a limit. "The differentiation of equations
consists simply in finding the limits of the ratio of finite differences of
110 Ibid., p. xxix.
111 Ibid., p. ix.
m Cajori, "Grafting of the Theory of Limits on the Calculus of Leibniz."
m Article, "Limite," in Encyclopedic Cf. also Pierpont, "Mathematical Rigor," p. 33.
248 The Period of Indecision
two variables included in the equation." 114 This D'Alembert believed
to be the true metaphysics of the calculus, admitting, incidentally,
that this was more difficult to develop than were the rules of appli-
cation.
Nieuwentijdt had criticized Leibniz's differentials of higher order as
nonexistant. D'Alembert, however, said that the distinction is irrele-
vant, for the differential notation is to be considered merely as a
convenient abridgment or manner of speaking, used to avoid the
circumlocution necessary in expressing the limit concept. He held that
no such thing as an infinitesimal existed in its own right.
A quantity is something or nothing: if it is something, it has not yet van-
ished; if it is nothing, it has literally vanished. The supposition that there
is an intermediate state between these two is a chimera. 115
Just as the calculus of first order differences was explained in terms
of limits, so D'Alembert defined infinitesimals of second and higher
orders in a terminology equivalent to that of limiting ratios. His
first explanation of these quantities lapsed into a phraseology danger-
ously resembling the naive assertion of Leibniz that dx 2 is defined as
being to dx as dx is to one, for D'Alembert said: "When one says
that a quantity is infinitely small with respect to a quantity which is
itself already infinitely small, this signifies merely that the ratio of
the first of these quantities to the second is always as much smaller
than the second quantity as is the latter than a given quantity."
He added immediately, however, the explanation interpretable in
terms of limiting ratios, "and that the ratio can be supposed as small
as we please in imagining the second quantity sufficiently small." 116
This definition of infinitesimals of higher order indicates the general
manner in which these quantities are interpreted at the present time,
but it lacks clarity and resolution. Furthermore, it does not un-
equivocally state that the limit of the ratio must be zero for infinites-
imals of higher order. These weaknesses in D'Alembert's explanation
were corrected early in the following century by Cauchy. Meanwhile
the interpretation of the infinitely small as a fixed infinitesimal con-
tinued in spite of D'Alembert's opposition.
"< Article, "Diflferentiel," p. 977.
w Melanges de litUrature, d'histoire, et de philosophic, pp. 249-50. m Ibid.
The Period of Indecision 249
As D'Alembert had interpreted the word infinitesimal as meaning
indefinitely small and had defined this in terms of limits, so also did
he try to clarify the other concept which had bothered mathematicians
since the Greek period — that of the infinite. He asserted— contrary to
the view of Fontenelle — that the notion of infinity is really that of the
indefinitely large and is only a convenient abridgment for the inter-
pretation in terms of the doctrine of limits. With this understanding,
he pointed out that one can have orders of infinitely large quantities
analogous to those of infinitesimals. A line is said to be infinite of the
second order with respect to another if the ratio of these is greater
than any given number; it is of the third order if the ratio of the
product of it by any finite number to the square constructed on the
other line is greater than any given number. 117
This interpretation of infinity is, of course, that which Euler had
in mind when he said that the logarithm of an infinite number is an
infinite number of lower order than that of any root of an infinite
number. 118 It corresponds to that which mathematicians use when
they speak of orders of infinity with respect to functions and is con-
cerned only with the limits of ratios. It has nothing to do with the
doctrine of infinite aggregates, which appeared in the late nineteenth
century and which was necessary rigorously to establish the calculus.
D'Alembert (like most of his contemporaries) would have been in-
capable of appreciating the modern notion of the actual infinite.
In geometry, D'Alembert explained, there was no need to suppose
the existence of an actual infinite, so that the question of its existence
did not concern mathematics. Whereas the modern concept of the
infinite is based upon arithmetic conceptions, D'Alembert 's inter-
pretations were largely in terms of geometry, even though in the
chart of the divisions of knowledge, in the preface of the Encyclopedic,
infinitesimal considerations are entered under, algebra rather than
geometry.
Because of his geometrical ideology, D'Alembert's elaboration of the
limit concept lacked the clear-cut phraseology necessary to make it
acceptable as a substitute for the infinitesimal interpretation. Thus to
say with D'Alembert that the secant becomes the tangent when the
two points are one and that it is therefore the limit of the secant, 119
UT Ibid., pp. 242 ff. "" Opera omnia, XV, 300. "» Melanges, V, 245-46.
250 The Period of Indecision
imposes the necessity of visualizing the process by which two points
become one, thus leaving the interpretation open to Zeno's criticisms.
To some mathematicians, at least, D'Alembert's limit concept
appeared to be enmeshed in as dark a metaphysics as was that of the
infinitely small. Consequently the majority of textbooks on the cal-
culus published on the Continent at that time continued to prefer
the explanations of Leibniz. Of twenty-eight publications appearing
from 1754 to 1784, fifteen interpreted the calculus in Leibnizian
terminology, six in terms of limits, four in terms of Euler's zeros, two
in terms of fluxions, and one (that of Lagrange) in terms of a method
to be described later. 120 Nevertheless, the limit idea continued from
time to time to be put forth as the logical manner of interpreting the
calculus. Hyacinth Sigismund Gerdil, for example, in 1760-61 121 fol-
lowed D'Alembert in saying that the infinite and the infinitesimal are
to be excluded from the calculus, their place being taken by limits. 122
Another supporter was found in A. G. Kastner, the author of a
popular textbook, Anfangsgrunde der Analysis des Unendlichen, which
appeared in 1761. In this Kastner said that he made use of Newton's
method of first and last ratios, although he availed himself also of the
abridged notation which the differential calculus afforded and which
he compared, as a sort of ellipsis, with the figures of speech used by
poets. 123 Following D'Alembert, he denied the existence of both the
infinitely large and the infinitesimal, although the latter insinuated
itself into his work nevertheless. In determining differentials, for
example, Kastner allowed the independent variable z to be given an
increment e, the function Z of z thereby taking on the increment E.
"If now e is indefinitely diminished, the limit which the ratio E : e
approaches indefinitely is called the ratio of the differentials of Z and
z and the infinitely small quantities e and E are called the differentials
of z and Z." 124 This lack of that distinction between the increments
and the differentials which is essential in the interpretation of the
calculus in terms of the derivative, shows clearly how little even pro-
fessed proponents of the limit concept could avoid using the infinitely
small in their explanations. Evidently Kastner did not recognize the
120 Cajori, "Grafting of the Theory of Limits on the Calculus of Leibniz."
m Cf. Lagrange, CEuvres, X, 269-70; VII, 598.
m See Moritz Cantor, Vorlesungen, IV, 643-44.
123 Anfangsgrunde, Vorrede, pp. xiii-xiv. m Anfangsgrunde, p. 10.
The Period of Indecision 251
significance of the admonition of Newton and D'Alembert that the
ultimate ratio was not a ratio of ultimate quantities.
That the differentials, rather than the differential quotient, were
fundamental in his thought is brought out in another connection. In
determining the differential E of Z = -^r t at the point for which
a
z = 0, we should first determine the derivative at this point and
then multiply this by e, the differential of z. The result is £ = 0.
Kastner, however, first found the increment E in Z, corresponding
to the increment e in z, and then substituted 2 = before determining
E e n
the limit of the ratio — . His result was E = -^zr v a conclusion which
e a
Euler and James Bernoulli had reached by their infinitesimal
methods. 128
A number of other mathematicians of the time felt that the infinitely
small could not furnish a satisfactory basis for the calculus. 126 Most
of these attempted to substitute some form of the limiting idea, but
there was one notable exception. Joseph Louis Lagrange displayed a
skeptical attitude toward the infinitely small, echoing Bishop Berkeley
in saying that the correctness of the results of the differential calculus
was to be explained as due to a compensation of errors. 127 However,
his attitude toward the limit concept was cool also, for he considered
this involved with metaphysical difficulties. 128 He felt that D'Alem-
bert's definition of the tangent as the limit of a secant was unsatis-
factory, inasmuch as after the secant has become the tangent, nothing
prevents it from continuing as a secant on the other side of the point
in question. 129
Furthermore, the method of fluxions did not appeal to Lagrange
because of the introduction of the irrelevant notion of motion. Euler's
presentation of dx and dy as failed also to satisfy him because he
felt that we have no clear and precise idea of the ratio of two terms
which become zero. 130 Lagrange, as a result, sought a simple algebraic
m Ibid., p. 13. See also James Bernoulli, Opera omnia, II, 1097; Euler, Opera omnia, X,
561.
m See Moritz Cantor, Vorlestmgen, IV, Section XXVI.
127 Lagrange, (Euvres, VII, 598. See also IX, 17.
128 (Euvres, VII, 325-26; cf. also III, 443, and IX, 18.
«• (Euvres, VII, 325. m (Euvres, IX, 17-18.
252 The Period of Indecision
method free from the objections found in others. As early as 1759 he
seems to have been satisfied that he had found this, for he wrote to
Euler in that year that he believed he had developed the true meta-
physics of the principles of mechanics and of the differential and
integral calculus, as far as was possible. 131 Lagrange probably had in
mind the method 132 which he proposed in a paper of 1772, "Sur une
nouvelle espece de calcul relatif a la differentiation et a Integration
des quantites variables." 133
Here he recalled that Leibniz had pointed out the analogy — now
called Leibniz's Rule — between differentials of all orders of the prod-
uct of two variables and the powers of the same order of a binomial
in these variables, and had also remarked that the same correspond-
ence subsisted between the negative powers and the integrals. Follow-
ing this suggestion, Lagrange made use of a similar analogy in con-
nection with infinite series. The series f(x -f h) = f(x) + f{x)h +
h 2
/""(a;)— + . . . had been known at least from the time of Taylor,
whose name it bears. In this series, the coefficients of the powers of h
involve the ratios of differentials, or of fluxions. However, the series
can be derived without reference to these notions. What would be
more natural, therefore, than to define differentials and fluxions in
terms of the coefficients of such a series? This procedure would (only
on the surface, as we know now) obviate the necessity of introducing
either limits or infinitesimals into the work, and the calculus would
thus be reduced to simple algebraic operations. This notion of the
differential and the integral calculus appeared to Lagrange to be
the clearest and simplest which had yet been given. It is, as one sees,
independent of all metaphysics and of any theory of infinitely small or
vanishing quantities. 134
Lagrange proceeded therefore to make Taylor's series fundamental in
his work, assuming implicitly that all functions allow of such an
expansion. The coefficients p, p', . . . g, q' , . . . , of the powers of h,
k, ... in the Taylor's series for u(x -f h, y -f- k, . . .) he defined as
the "fonctions derivees" of u. The differential calculus then consisted,
m (Euvres, XIV, 173; cf. Ill, 443, and VII, 325-28.
132 See Jourdain, "The Ideas of the 'Fonctions analytiques' in Lagrange's Early Work.'*-
m See (Euvres, III, 439-76. u« (Euvres, III, 443.
The Period of Indecision 253
for Lagrange, in "finding directly and by simple and facile procedures
the derived functions p,p',..., q, q', - ■ • of the function «"; and the
integral calculus consisted, inversely, in "determining by means of
these latter functions the function «." 135
Lagrange's method is based on the unwarranted supposition that
every function can be so represented and handled. Moreover, the
escape from the infinitely large and the infinitely small, as well as
from the limit concept, is only illusory, inasmuch as these notions
enter into the critical question of convergence which Lagrange did
not adequately consider. Furthermore, his method lacks the opera-
tional suggestiveness and facility which the Leibnizian ideas and
notation afforded. Logically, however, the Lagrangian definition had
an advantage in that it sought, as had the work of Euler, to make
fundamental the formalism of the theory of functions, rather than the
preconceptions of geometry, mechanics, or philosophy.
Lagrange has been criticized 136 for giving up, in favor of mathe-
matical formalism, the "generative" concept which has frequently
been felt to be the basis of the methods of fluxions and differentials.
Such criticism is based on a failure to recognize that mathematics is
most useful when unencumbered by psychological preconceptions.
Euler had attempted misdirectedly to formalize the Leibnizian con-
ceptions by making the differentials zeros. D'Alembert had made an
effort to present a satisfactory idea of the notion of limit, but had
failed to give the concept a clear and precise formalism which would
make it logically unequivocal. Lagrange, therefore, sought still
another mode of presentation, based on the function concept which
Euler had emphasized and popularized. Incidentally, in so doing he
focused attention for almost the first time upon the quantity which
is now the central conception in the calculus — that of the derived
function, or the derivative, or the differential coefficient (a termin-
ology reminiscent of Lagrange's method, which was, however, intro-
duced later by Lacroix). 137 Lagrange, in this connection gave not only
the name from which the word derivative was adopted, but also the
notation f'x, modifications of which are still conveniently used.
138 Ibid.
m Cohen, Das Princip der Infinilesimalmethode und seine Geschichte, p. 100.
™ See Bibliotheca Mathematica (3), I (1900), 517 for notes on the origin of these desig-
nations.
254 The Period of Indecision
Newton had emphasized the method of prime and ultimate ratios
of increments or of fluxions. Although this ratio can be interpreted as
a single number or quantity, which we now call the derivative, Newton
seems to have had in mind the idea of this quantity as the ratio of
increments or of fluxions, or of quantities proportional to these.
Moreover, although the fluxions themselves can only be defined
rigorously as derivatives, Newton does not appear to have realized
this.
In the differential calculus likewise it is clear that although Leibniz
realized the significance of the ratio of two infinitesimals, he never
seems to have thought of this ratio as a single number, but rather as
a quotient of inassignables, or of assignables proportional to these.
With D'Alembert's insistence that the differential calculus was to be
rigorously interpreted only in terms of limits, one comes close to the
conception of a derivative, but even here there is the lack of the
notion of a single function, or number, obtained as the limit of a
single infinite sequence.
Like Newton and Leibniz, D'Alembert appears to have had in
mind not a function, but two sides of an equation, the limits of which
are equal. Kastner likewise displayed the predominance of the idea
of a ratio. Although he recognized the fundamental significance of the
limit concept, he nevertheless defined the differential quotient literally
as a ratio of differentials. 138 In Lagrange's method the term derivative
can for the first time be applied with strict propriety, for his "fonction
derivee" is merely a single coefficient of a term in an infinite series
and is completely divested of any idea of ratio, or limiting equality.
It is a single quantity or function, and although Lagrange's definition
was not that which was to be accepted in the end, the notion of a
derived function may well have aided in making the general accept-
ance of the current definition possible.
Although Lagrange's method appeared in the Miscellanea Taurin-
ensia for 1772, it received little recognition — perhaps as a result of the
novelty of the ideas and the notation involved — and the search for a
satisfactory basis for the calculus continued. To encourage efforts
in this direction the Berlin Academy, of which Lagrange at the time
was president, in 1784 offered a prize for the best exposition of a clear
138 Anfangsgriinde, p. 4.
The Period of Indecision %55
and precise theory of the mathematical infinity. The prize-winning
essay was that of Simon L'Huilier: Exposition elementaire des prin-
cipes des calculs superieurs. This was published in 1787 and appeared
again (in Latin) in 1795. In this work L'Huilier proposed to show
that "the method of the ancients, known under the name of Method
of Exhaustion, conveniently extended, suffices to establish with cer-
tainty the principles of the new calculus." 139 In conformity with this
purpose, he modified the method of exhaustion to interpret it in terms
of limits. Making the limit concept basic in his exposition, L'Huilier
agreed with D'Alembert that "in the differential calculus it was not
necessary to pronounce the name differential quantity." 140
As in modern textbooks, L'Huilier made the differential ratio or
quotient fundamental, defining this as the limit of the ratio of the
increment in the function to that in the independent variable.
L'Huilier regarded this form of presentation of the calculus as a
development of that of Newton and other English authors, but his
exposition indicated an advance over their work. He focused atten-
tion upon a single number (the derivative) as the limit of a single
variable (the ratio of the increments), rather than upon an ultimate
ratio of two evanescent quantities, or of two fluxions, or of any two
quantities which have the same ratio as these. Although he retained
dy
the name "differential quotient" and the symbol — to represent this
ax
quantity, he insisted that the latter was nothing but a symbol which
was to be interpreted as a single number. 141 In dealing with differential
d 2 y
quotients of higher orders he again warned that the symbol — was
not to be broken up as a quotient. This is in striking contrast with the
work of Newton and Leibniz, in which fluxions and differentials of
any order were regarded as having a significance independent of the
ratio or equation in which they entered. The differential quotient of
L'Huilier was a single number or function, equivalent to Lagrange's
derived function, and represented essentially the present conception
of the derivative.
Although L'Huilier 's definition of the differential quotient is in
139 Exposition iUmentaire, p. 6.
"° Ibid., p. 141. m IMd., p. 32.
%56 The Period of Indecision
most respects that to be found in present day elementary textbooks
on the calculus, he does not seem to have been aware of the possible
difficulties involved in the limit concept. He had avoided the mysti-
cism of the infinitesimal, the vagueness of the ultimate ratio, and the
inanity of the symbol -; but he failed to appreciate that the subtlety
of the limit concept was to make an extremely careful definition
essential. L'Huilier was dealing only with very simple functions, so
that he was unaware of the inadequacies of his presentation. His
variable was always less than or greater than his limit. "Given a
variable quantity always smaller or greater than a proposed constant
quantity; but which can differ from the latter by less than any pro-
posed quantity however small; this constant quantity is called the
limit in greatness or in smallness of the variable quantity." 142 The
variable could not oscillate, as it may under our more general view.
More seriously, L'Huilier fell into an error which is suggested by
the vague idea of uniformity expressed in the law of continuity of
Leibniz. He said that "if a variable quantity at all stages enjoys a
certain property, its limit will enjoy this same property." 143 That
this notion persisted also in the nineteenth century is apparent from
the statement of William Whewell: "The axiom . . . that what is
true up to the limit is true at the limit, is involved in the very con-
ception of a limit." 143a The falsity of this doctrine is immediately
apparent from the fact that irrational numbers may easily be
defined as the limits of sequences of rational numbers, or from
the observation that the properties of a polygon inscribed in a
circle are not those of the limiting figure — the circle. The mistake
of L'Huilier in this connection was probably the result of his
failure to see that the limit concept was to be identified with
the nature of infinite converging sequences and with the question
of the nature of real numbers and the continuum. The inadequacy of his
conception of number may be further inferred from the fact that he felt
it necessary to distinguish between limiting values and limiting ratios.
10 Ibid., p. 7.
143 Exposition ilimentaire, p. 167. Cajori ("Grafting of the Theory of Limits on the Calcu-
lus of Leibniz"), apparently unaware of this statement, has said that this principle was used
but not stated in the eighteenth century.
143 * History of Scientific Ideas, I., 152.
The Period of Indecision 257
Although L'Huilier correctly sought the basis of the calculus in
the limit concept, his exposition of this was an oversimplification of
what was later realized to be a very difficult question. Following
D'Alembert in regarding the infinite from the point of view of mag-
nitude rather than of aggregation, he denied the existence of an
actually infinite quantity because he felt that its acceptance would
lead to such "contradictions" as t» + n = °° - n. He consequently
maintained that he had shown the calculus to be independent of all
idea of the infinite, whether large or small. In this he failed to realize
that the whole theory of limits is based, in the last analysis, upon
that of infinite aggregates. This fact was not clearly recognized until
the following century.
L'Huilier's monograph was not widely read, nor were his views
generally accepted at the time. The indecision as to the true basis of
the calculus remained as acute as before. As a result, there appeared
in 1797 what was perhaps the most famous attempt to clear up the
difficulties in the situation: the Reflexions sur la metaphysique du
calcul infinitesimal, by L. N. M. Carnot, the remarkable soldier,
administrator, and mathematician to whom the French Assembly
gave the title "L'Organizateur de la Victoire." Carnot's work enjoyed
a truly remarkable popularity, appearing in numerous editions and
several languages from that time until quite recently. 144
In view of the lack of clarity and uniformity in the then-current
expositions of the calculus, Carnot wished to make the theory rigidly
precise. Considering the many conflicting interpretations of the sub-
ject, he sought to know "in what the veritable spirit of the infinitesimal
analysis consisted." 145 In his selection of the unifying principle, how-
ever, he made a most deplorable choice. He concluded that "the true
metaphysical principles of the Infinitesimal Analysis . . . are never-
theless . . . the principles of the compensation of errors," 146 as Berk-
144 The first edition of Paris, 1797, was followed by an enlarged second edition at the
same place in 1813. Unless otherwise stated, references in this work are to this second
edition. The first edition was translated into English by W. Dickson in the Philosophical
Magazine, VIII (1800), 222-40; 335-52; IX (1801), 39-56. The second edition appeared
in an English translation by W. R. Browell at Oxford, in 1832, as Reflexions on the Meta-
physical Principles of the Infinitesimal Analysis. Other French editions appeared in Paris
in 1839, 1860, 1881, and 1921 (2 vols.) A Portuguese translation appeared at Lisbon in
1798, a German one at Frankfurt a. M. in 1800, and an Italian one at Pavia in 1803.
146 Reflexions sur la mitaphysique dn calcul infinitesimal, p. 1. 146 Browell trans., p. 44.
258 The Period of Indecision
eley and Lagrange had suggested. In his expansion of this view, he
reverted substantially to ideas which Leibniz had expressed. He held
that to be certain that two designated quantities are rigorously equal,
it is sufficient to prove that their difference cannot be a "quantite
designee." 147 Paraphrasing Leibniz, Carnot said further: that for any
quantity one may substitute another which differs from it by an
infinitesimal; 148 that the method of infinitesimals is nothing more than
that of exhaustion reduced to an algorithm; 149 that "quantites inap-
preciables" are merely auxiliaries which are introduced, like imaginary
numbers, only to facilitate the computation, and which are eliminated
in reaching the final result. 150
Carnot even echoed the favorite explanation of Leibniz in terms of
the law of continuity. He held that one can envisage the infinitesimal
analysis under two points of view, according as the infinitesimals are
taken to be "quantites effectives" or are regarded as "quantites
absolument nulles." (Leibniz, however, had not admitted them as
absolutely, but only relatively, zero.) In the first case, he felt that the
calculus was to be explained upon the basis of a compensation of
errors: "imperfect equations" were to be made "perfectly exact" by
the simple expedient of eliminating the quantities whose presence
occasioned the errors; 151 in the latter case he considered the calculus
an "art" of comparing vanishing quantities with each other in order
to discover from these comparisons the relationships between the pro-
posed quantities. 152 To the objection that these vanishing quantities
either are or are not zero, Carnot responded that "what are called
infinitely small quantities are not simply any null quantities at all,
but rather null quantities assigned by a law of continuity which
determines the relationship." 153 This explanation is strikingly like that
given by Leibniz about a century earlier.
Along with his establishment of the "true metaphysics" of the
infinitesimal calculus, Carnot proceeded to show that the diverse
views of the subject were essentially reducible to this same basis.
147 Riflexions sur la mitapkysique du calcul infinitesimal, p. 31.
148 /Wrf., p. 35. i« Ioid} p> 39
160 Ibid., pp. 38-39. "Les mathematiques ne sont-elles pas remplies de pareilles enigmes?"
said Carnot, in this connection.
151 Cf. Dickson's translation, p. 336.
162 Riflexions sur la mitapkysique du calcul infinitesimal, p. 185. 1M Ibid., p. 190.
The Period of Indecision 259
In demonstrating this, he pointed out that the method of exhaustion
made use of analogous systems of known auxiliary quantities. Newton's
method of prime and ultimate ratios was similar, except that in it the
lemmas freed the work from the need for the argument by a double
reductio ad absurdum. Carnot felt that the methods of Cavalieri and
Roberval also were admittedly corollaries of the method of exhaus-
tion. 154 Descartes' method of undetermined coefficients, he held,
touched the infinitesimal analysis closely, the latter being but "a
felicitous application" of the former! 155 The method of limits he recog-
nized to be not different from that of first and last ratios, so that it
was likewise a simplification of the method of exhaustion. 156 Further-
more, these methods all lead to the results of infinitesimal analysis,
but by a difficult and circuitous route. 157 Lagrange's method he like-
wise saw linked to the infinitesimal calculus, in that it neglects the
other terms of the infinite series. 158 The divers points of view he
therefore felt, somewhat as had L'Hurlier, to be merely simplifications
of the method of exhaustion which were effected by reducing this to a
convenient algorithm. Inasmuch as the infinitesimal method combined
the facility of the procedures of approximate calculation with the
exactitude of the results of ordinary analysis, he saw no point in
attempting, under the guise of greater rigor, to substitute for it any
less natural method. 159
Although Carnot's work was widely read, it can hardly be said to
have led to a clearer understanding of the difficulties inherent in the
new analysis. Although he realized that differentials were to be denned
somewhat as variables, anticipating, to a certain extent, the view of
Cauchy, he was unable to give suitable definitions to these because,
like Leibniz, he thought in terms of equations rather than of the
function concept which led Cauchy to make the derivative funda-
mental. Furthermore, Carnot, one of a school of mathematicians who
emphasized the relationship of mathematics to scientific practice, 160
appears, in spite of the title of his work, to have been more concerned
about the facility of application of the rules of procedure than about
the logical reasoning involved. In this respect his work resembles
154 Ibid., pp. 139-40. I56 Ibid., pp. 150-51. 1M Ibid., p. 171.
157 Ibid., p. 192. i» Ibid., pp. 194-97. » Ibid., pp. 215-16.
160 Merz, A History of European Thought in the Nineteenth Century, II, 100-1.
260 The Period of Indecision
that of Leibniz, whose explanations of differentials he so largely
paraphrased, and whose method he defended with almost polemic
warmth.
Mathematicians on the Continent had been in essential agreement
that pragmatically the differential calculus offered the best method
of procedure; but it was precisely on the logic of the matter that
they were at variance. On the latter point Carnot was of little assist-
ance, for after pointing out what most men had long realized — that
all of the methods of the new analysis were essentially related — he
proceeded to make basic the infinitesimal system, the one which was
logically, perhaps, the weakest of all. In so doing he pointed toward
a view diametrically opposed to that which D'Alembert 161 and
L'Huilier had indicated and along which the rigorous development
was ultimately to proceed, following the work of Cauchy. 162
The year 1797, in which the first edition of Carnot's work was
published, saw the appearance also of the famous work of Lagrange,
Theorie des fonctions analytiques, contenant les principes du calcul
differentiel, degages de toute consideration d'infiniment petits ou d'evan-
ouissans, de limites ou de fluxions, et redmts a V analyse algebrique des
quantites finies. This book developed with care and completeness the
characteristic definition and method in terms of "fonctions derivees,"
based upon Taylor's series, which Lagrange had proposed in 1772.
In it the author gave not only an attempted proof of the incorrect
theorem that every continuous function may be so expanded, but
also the determination of the "fonctions derivees" (or derivatives) of
the elementary functions, and numerous applications to geometry and
mechanics. Carried along by the authority of Lagrange's great repu-
tation, the method now enjoyed a short-lived period of comparative
success. As had been the case with Maclaurin's treatise on fluxions,
mathematicians praised the new method highly, 163 although they
seldom used it. They explained that inasmuch as the notations were
181 Mansion (Esquisse de I'kistoire du calcul infinitesimal, p. 290, n.) has misdirectedly
characterized Carnot's ideas as a development of those of D'Alembert.
1,2 The assertion (See Smith, "Lazare Nicholas Marguerite Carnot," p. 189) that Carnot
"paved the way for Cauchy's notable memoir" is justified only in the most general sense.
143 Cf. the review in Monthly Review, London, N. S., XXVIII (1799), appendix, pp. 481-
99; also Valperga-Caluso, "Sul paragone del calcolo delle funzioni derivate coi metodi
anteriori"; also the set of monographs by Froberg, and others, called De analytica calculi
differentialis et integraiis theoria, inventa a eel. La Grange.
The Period of Indecision 261
less convenient and the calculations involved more embarrassing than
those found in the methods of the ordinary differential and integral
calculus, it was sufficient that one were assured through Lagrange's
method of the legitimacy of other more expeditious methods. This
seems to have been the view of Lagrange himself, for side by side
with his method of derived functions he continued throughout his life
to employ the notation of differentials.
Most of the objections to the method of Lagrange were based upon
the inconveniences of the notation and the operations, but before long
doubts began to arise as to the correctness of the principle that all
continuous functions could be expanded in Taylor's series. It was
pointed out 164 that such an expansion was possible only for the more
simple functions, and that consequently the method was of limited
applicability. Furthermore, Sniadecki, a Pole, correctly explained that
the method was fundamentally identical with the limit method. 165
However, the views of convergence, continuity, and function held at
the time were not sufficiently definite to permit a deeper clarification
of these ideas.
An interesting but somewhat misdirected tirade against Lagrange's
point of view was delivered by another Polish mathematician, Hoene
Wronski. An eager devotee of the differential method of Leibniz and
of the transcendental philosophy of Kant, he protested with some
asperity against the ban on the infinite in analysis which Lagrange
had wished to impose. He criticized Lagrange not so much for the
absence of logical rigor in his free manipulation of infinite series —
although he did pertinently ask where Lagrange got the series f(x + i)
= A + Bi + Ci 2 + Di 3 -f . . . with which he opened his proof of
Taylor's series — as for his lack of a sufficiently broad view. Wronski
believed that modern mathematics is to be based on the "supreme
algorithmic law" Fx ~ A Qo + ^4iS2i + A& 2 + A 3 Q, 3 -f . . . , where
the quantities fio, Qi, fl2, 83, • • . are any functions of the variable x.
Being the supreme law of mathematics, the irrecusable truth of this
law he held to be not mathematically derived, but given by trans-
cendental philosophy. 166
164 See Dickstein, "Zur Geschichte der Prinzipien der Infinitesimalrechnung"; cf. also
Lacroix, Traiti du calcid, 2d ed., Ill, 629-30.
166 Dickstein, "Zur Geschichte der Prinzipien der Infinitesimalrechnung."
1,8 Refutation de la theorie des fonctions analytiques, Vol. IV of Wronski's (Euvres.
262 The Period of Indecision
Wronski was, of course, correct in saying that the method of
derived functions was too restricted, in that it was limited only to
certain functions which could be so expanded. However, his general
views on the calculus were far from those accepted at the present
time. Whereas Lagrange had attempted to give a formal logical
justification of the subject, Wronski asserted that the differential
calculus constituted a primitive algorithm governing the generation of
quantities, rather than the laws of quantities already formed. Its
propositions he held to be expressions of an absolute truth, and the
deduction of its principles he consequently regarded as beyond the
sphere of mathematics. The explication of the calculus by the methods
of limits, of ultimate ratios, of vanishing quantities, of the theory of
functions, he felt constituted but an indirect approach which pro-
ceeded from a false view of the new analysis. In seeking to examine
the principles of the subject by purely mathematical means, Wronski
believed geometers were simply wasting time and effort. He called
upon them to give up "that servile imitation of ancient geometers in
avoiding the infinite and that which depends upon it." 167
In his own work Wronski made a more uncritical use of the infinite
than had even Wallis and Fontenelle. He said, for example, that "the
absolute meaning of the number t was given by the expression . . .
4 oo f ]
^r== J (1 + V - i)i - (l _ V - i)i I ."i68 Perhaps because of the
novelty of his notation, as well as of this bizarre use of the symbol <*> ,
Wronski did not exert a strong influence upon the development of
the calculus. The mathematics of the time was about to accept what
he opposed: the rigorous logical establishment of the calculus upon the
limit concept. Nevertheless, the work of Wronski represents an ex-
treme example of a view which we shall find recurring throughout the
nineteenth century. In regarding the calculus as a means of explaining
the growth of magnitudes, followers of this school of thought were
to attempt to retain the concept of the infinitely small, not as an
extensive quantity but as an intensive magnitude. Mathematics has
excluded the fixed infinitely small because it has failed to establish
the notion logically; but transcendental philosophy has sought to
preserve primitive intuition in this respect by interpreting it as
w Ibid., pp. 39-40. "• Ibid., p. 69.
The Period of Indecision 263
having an a priori metaphysical reality associated with the generation
of magnitude.
Lagrange's Theorie des fonctions was only one, but by far the most
important, of many attempts made about this time to furnish the cal-
culus with a basis which would logically modify or supplant those
given in terms of limits and infinitesimals. Condorcet, Arbogast,
Servois, and others put forth methods similar to that of Lagrange. 169
Condorcet, as early as 1786, had begun a work on the calculus, based
on series and finite differences, but this was interrupted by the Revolu-
tion and did not appear. 170
In 1789 L. F. A. Arbogast presented to the Academie des Sciences a
"true theory of the differential calculus" along the lines of Lagrange
and Condorcet. This work, published in 1800 under the title Du
calcul des derivations, sought to establish the principles of the calculus
independently of limits and the infinitely small, and with the sim-
plicity and certitude found in ordinary algebra. 171 Arbogast assumed,
as had Lagrange, that the function F( a + x) could be expanded in a
series of powers of x, and then showed that the coefficients in this
could be identified with the more familiar differential quotient. 172
These new methods resemble the attempts which had been made
in England, following the Berkeley- Jurin-Robins dispute, to give
arithmetical procedures for the calculus. They serve to indicate, as
did the British controversy, a lack of satisfaction, at the time, with
the methods of limits, fluxions, and infinitesimals. The full title of
Lagrange's Theorie des fonctions indicates his discontent with these
methods. F. J. Servois, in his Essai sur un nouveau mode d 'exposition
des principes du calcul differentiel, which appeared in 1814, 173 expressed
himself more strongly. He called the method of limits a "gothique
hypothese" 174 and that of differentials a "strabisme infinitesimal." 176
IM For references to this work, see Dickstein, op. cit.; Moritz Cantor, Vor lesungen, IV,
Section XXVI; and Lacroix, Traitt du calcul differential et du calcul integral, I, 237-48,
and preface.
170 See Lacroix, Traits du calcul, I, xxii-xxvi.
171 See Zimmermann, A rbogast als Mathematiker und Historiker der Mathematik, pp. 44-45 ;
cf. also Arbogast, Du calcul des derivations, Preface; and Lacroix, TraiU du calcul, Preface.
172 Du calcul des derivations, pp. xii-xiv; cf. also p. 2.
173 This appeared in the Annates de Mathematiques, V (1814-15), pp. 93-141, and was
also published separately in Nimes, 1814. References given in this work are to the Nlmes
publication.
»< Essai, p. 56. ire /&#., p , 65.
264 The Period of Indecision
The introduction of the idea of infinity he objected to as useless; and,
with reference to the views which Wronski had expressed on the cal-
culus in his Refutation of Lagrange, he asserted that he "had well
foreseen, on reading Kant, that geometers would sooner or later be
the object of the cavils of his sect." 176 Servois tried, in his turn, to
establish the calculus on a combination of finite differences and
infinite series, in which the idea of limits was, as in the case of the
method of Lagrange, ostensibly eliminated by a disregard of the
essential question of convergence.
Further attempts along this line continued for sometime: in 1821
in Eine neue Method efiir den Infinitesimalkalkul of Georg von Buquoy ;
in 1849 in the Essai sur la metaphysique du calcul integral of C. A.
Agardh; and as late as 1873 in an article, "Exposition nouvelle des
principes de calcul differentiel," by J. B. Brasseur. 177 These invariably
depended upon expansions in series to avoid "any metaphysical no-
tions," and in this respect they resembled closely the earlier efforts of
Lagrange, Condorcet, Arbogast, and Servois. However, at almost pre-
cisely the period of the work of Servois a tendency toward rigor in
series was setting in. This made clear the basic weakness of all such
attempts to avoid the method of limits through an uncritical formal
manipulation of infinite series. The final formulation of the calculus
was not destined to be in terms of one of these new methods which
were developed, but was to be based on one of the very notions they
sought to avoid — that of a limit.
In the year 1797, in which Carnot attempted a concordance of all
the systems of the calculus and in which Lagrange tried seriously to
establish his method, there was published the first volume of perhaps
the most famous and ambitious textbook on the subject which had
appeared up to that time — Traite du calcul differentiel et du calcul
integral of Lacroix. Lacroix declared in the preface of this work that
it had been the new method of Lagrange which had inspired him to
compose a treatise on the calculus which should have as a basis the
luminous ideas which this method had substituted for the infinitely
small. His text, however, indicates well the indecision of the period,
for in spite of the professed aim of the work, the foundation of the
subject remained somewhat in doubt. Lacroix interpreted the method
174 Ibid., p. 68. m See Bibliography below, for full citations of these works.
The Period of Indecision 265
of series of Lagrange in terms of the limits of D'Alembert and L'Huilier.
In this respect, however, he obscured the significance of this relation-
ship by speaking of the limit of divergent series, and by following
Euler in the study of such infinite series asl — 1+2 — 6-f-24 —
120 + . . . m Furthermore, he did not share with Lagrange the sus-
picion that the method of Leibniz was based on a false idea of the
infinitely small. Lacroix admittedly made use of infinitesimals. 179
Although he accepted the metaphysics of Lagrange, he adopted the
differential notation of Leibniz. 180 This fact at times confused his
thought and led him, as it had Euler, to regard the differential coeffi-
cient (as Lacroix called it) as a quotient of zeros. 181
The lack on the part of Lacroix of critical distinction in dealing
with the methods of Newton, Leibniz, and Lagrange gave to his work
an appearance somewhat resembling the attempt of Carnot to demon-
strate the congruity of the numerous representations of the calculus.
The mathematician and astronomer Laplace praised this attitude on
the part of Lacroix, saying that such a rapprochement of methods
served as a mutual clarification, the true metaphysics probably being
found in what they had in common. 182 It appears, however, that at
that time at least it was deplorable, for it led to a confusion of thought
on the subject just when logical precision was most needed. However,
in his more popular work of 1802 — the Traite elementaire, an abridg-
ment of his larger treatise — Lacroix omitted the method of Lagrange
and made the explanation in terms of limits basic, although again
with a lack of rigor which made interpretations in terms of infinites-
imals possible. The success of this work, which went through many
editions (the ninth French edition appearing in 1881) and was trans-
lated into several languages, led to other texts of the same type; and
it was largely through these that the method of limits became familiar,
if not rigorous. It was through such texts that the Leibnizian notation
and the doctrine of limits supplanted in England the method of
fluxions and interpretations which had already become hopelessly
confused with the infinitely small.
The year 1816, in which Lacroix's shorter work was translated into
m Traite du calcul, HI, 389.
m Ibid., I, 242. "> Ibid., p. 243. ™ Ibid., I, 344.
m Letter to Lacroix, published in Le Nouveau Traits du calcul difftrentid, 2d ed., preface,
p.XLX.
266 The Period of Indecision
English, "marks an important period of transition," 183 because it
witnessed the triumph in England of the methods used on the Conti-
nent. This particular point in the history of mathematics marks a
new epoch for a far more significant reason, for in the very next year
there appeared a work by Bolzano — Rein analytischer Beweis des
Lehrsatzes dass zwischen je zwei . Wertlien, die ein entgegengesetztes
Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege —
which indicated the rise of the period of mathematical rigor in all
branches of the subject. 184 In the calculus the new attitude resulted
in the logical establishment of the higher analysis upon the limit
concept, and thus brought to an end the period of indecision which
had begun with the inventions of the method of fluxions and of the
differential calculus.
m Cajori, A History of the Conceptions of Limits and Fluxions, pp. 270-71.
m See Pierpont, "Mathematical Rigor," pp. 32-34.
VII. The Rigorous Formulation
THE OBJECTIONS raised in the eighteenth century to the
methods of fluxions, of prime and ultimate ratios, of limits, of
differentials, and of derived functions were in large measure un-
answered in terms of the conceptions of the time. The arguments were
in the last analysis equivalent to those which Zeno had raised well
over two thousand years previously and were based on questions of
infinity and continuity. The proponents of all methods except that of
differentials, however, protested that they had no need to invoke the
notion of the infinite and disregarded entirely that of continuity. The
advocates of the differential calculus, in turn, although attempting to
justify its procedures in terms of these concepts, were quite unable
to furnish logically consistent explanations of them. By most mathe-
maticians these were considered to be metaphysical ideas and so to lie
beyond the realm of mathematical definition.
It is interesting, in looking back, to see that the methods most
adverse to the introduction into mathematics of the notions of infinity
and of continuity were precisely those which made this introduction
possible. The method of limits, which at that time appeared to lead
to neither infinity nor the law of continuity, was to furnish the logical
basis for these; and the method of Lagrange, which was developed in
order to avoid these difficulties, was to bring up questions which
pointed the way toward their solution. We have seen that in the early
nineteenth century critics of Lagrange began to question the validity
of his principle that a continuous function can always be expressed,
by means of Taylor's theorem, as an infinite series. They began to ask
what was meant by a function in general and by a continuous function
in particular, and to criticize the almost indiscriminate use of infinite
series. Lagrange had made a start in the direction of greater care in
the use of series when he pointed out that one must consider the
remainder in every case. This warning serves to indicate, perhaps,
why he thought he had avoided the use of the infinite and the infi-
nitely small. Like Archimedes, he evidently did not consider the series
as continued to infinity, but only to a point at which the remainder
268 The Rigorous Formulation
was sufficiently small. In the nineteenth century, however, the con-
cept of infinity was to become basic in the calculus through the use
of infinite series and infinite aggregates.
One of the pioneers in the matter of greater rigor in the funda-
mental conceptions of the calculus — in its arithmetization and in the
careful study of the infinite — was the Bohemian priest, philosopher,
and mathematician, Bernhard Bolzano. 1 In 1799 Gauss had given a
proof of the fundamental theorem of algebra — that every rational,
integral algebraic equation has a root — using considerations from
geometry. Bolzano, however, wished a proof involving only con-
siderations derived from arithmetic, algebra, and analysis. As Lagrange
had felt that the introduction of time and motion into mathematics
was unnecessary, so Bolzano sought to avoid in his proofs any con-
siderations derived from spatial intuition. 2
This attitude made necessary, in the first place, a satisfactory
definition of continuity. The calculus may, indeed, be thought of as
having arisen from the Pythagorean recognition of the difficulty
involved in attempting to substitute numerical considerations for
supposedly continuous geometrical magnitudes. Newton had avoided
such embarrassment by appealing to the intuition of continuous
motion, and Leibniz had evaded the question by his appeal to the
postulate of continuity. Bolzano, however, gave a definition of con-
tinuous function which, for the first time, indicated clearly that the
basis of the idea of continuity was to be found in the limit concept.
He defined a function f(x) as continuous in an interval if for any
value of x in this interval the difference f(x + Ax) — f(x) becomes
and remains less than any given quantity for Ax sufficiently small,
whether positive or negative. 3 This definition is not essentially differ-
ent from that given a little later by Cauchy and is fundamental in the
calculus at the present time.
In presenting the elements of the calculus, Bolzano realized clearly
that the subject was to be explained in terms of limits of ratios of
1 See Stolz, "B. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung."
Although Bolzano was born and died at Prague, Quido Vetter ("The Development of
Mathematics in Bohemia," p. 54) has called him a Milanese, inasmuch as his father was
a native of northern Italy.
2 Rein analytiscker Beweis, pp. 9-10.
3 See Bolzano's Schriften, I, 14; cf. also Rein analytiscker Beweis, pp. 11-12.
The Rigorous Formulation 269
finite differences. He defined the derivative of F(x) for any value of
Fix -f- Ax) — F(x)
x as the quantity F'(x) which the ratio approaches
Ax
indefinitely closely, or as closely as we please, as Ax approaches
zero, whether Ax is positive or negative. 4 This definition is in essence
the same as that of L'Huilier, but Bolzano went further in explaining
the nature of the limit concept. Lagrange and other mathematicians
had felt that the limit notion was bound up with a quotient of evan-
dy
escent quantities or of zeros. Euler had explained — as a quotient of
ax
zeros, and in this respect Lacroix tended to follow him. Bolzano, how-
ever, emphasized the fact that this was not to be interpreted as a
ratio of dy to dx or as the quotient of zero divided by zero, but was
rather one symbol for a single function. 6 He held that a function has
no determined value at a point if it reduces to -. However, it may
have a limiting value as this point is approached, and he correctly
indicated that, by adopting the limiting value as the meaning of -,
the function may be made continuous at this point. 6
Ever since the invention of the calculus it had been felt that,
inasmuch as the subject was bound up with motion and the growth
of magnitudes, the continuity of a function was sufficient to assure
the existence of a derivative. In 1834, however, Bolzano gave an
example of a nondifferentiable continuous function. This is based upon
a fundamental operation which may be described as follows: Let
PQ be a line segment inclined to the horizontal. Let it be halved by
the point M, and subdivide the segments PM and M Q into four
equal parts, the points of division being Pi, Pi, ^3, and Q 1} Q 2 , Q 3 .
Let Pz be the reflection of P% in the horizontal through M, and let
Qz be the reflection of Q 3 in the horizontal through Q. Form the
broken line PP 3 'MQ 3 'Q. Now apply to each of the four segments of
this broken line the fundamental operation as described above, ob-
taining in this manner a broken line of 4 2 segments. Continuing
* Schriften, I, 80-81 ; cf . Paradoxien des Unendlichen, p. 66.
* Paradoxien des Unendlichen, p. 68. • Schriften, pp. 25-26.
270 The Rigorous Formulation
this process indefinitely, the broken-lined figure will converge toward
a curve representing a continuous function nowhere differentiable. 7
This illustration given by Bolzano might have served in mathe-
matics as the analogue of the experimentum cruris in science, showing
that continuous functions need not, in spite of the suggestions of
geometrical and physical intuition, possess derivatives. However,
because the work of Bolzano did not become known at the time, such
a role was reserved for the famous example of such a function given by
Weierstrass about a third of a century later. 8
Lagrange had held that his method of series avoided the necessity
of considering the infinitesimals or limits, but Bolzano pointed out
that in the case of infinite series it is necessary to consider questions
of convergence. These are analogous to limit considerations, as is
obvious from the statement of Bolzano that if the sequence Fi(x),
F 2 (x), F 3 (x), . . . , F n (x), . . . , F n + r (x), ... is such that the differ-
ence between F n (x) and F n + r (x) becomes and remains less than any
given quantity as n increases indefinitely, then there is one and only
one value to which the sequence approaches as closely as one pleases. 9
This fundamental proposition of Bolzano will be seen later to have
also a significance with respect to the general definition of real number
and the arithmetical continuum.
Bolzano felt, in spite of the paradoxes presented by the notions of
space and time, that any continuum was to be thought of as ulti-
mately composed of points. 10 His view in this respect resembles that
of Galileo, to whom he referred in this connection. 11 Although he
denied the existence of infinitely large and infinitely small magni-
tudes, he maintained, with Galileo, the possibility of an actual infinity
with respect to aggregation. He remarked, with respect to such
assemblages, the paradox which Galileo had pointed out: that the
part could in this case be put into one-to-one correspondence with the
whole. The numbers between and 5, for example, could be paired
with those between and 12. 12 Bolzano's views on the infinite are
substantially those which mathematicians have adopted since the
7 See Kowalewski, "Uber Bolzanos nichtdifferenzierbare stetige Function."
8 Waismann (Einfuhrung in das mathematische Denken, p. 122) mistakenly represents
Weierstrass as the first to give such an example.
9 Rein anaiytischer Beweis, p. 35. 10 Paradoxien des Unendlichen, pp. 75-76.
11 Ibid., pp. 89-92. u Ibid., pp. 28-29.
The Rigorous Formulation 271
time of Cantor, except that what Bolzano had considered as different
powers of infinity have been discovered to be of the same power.
Bolzano, however, sought to prove the existence of the infinite upon
theological grounds, whereas later in the century the property which
he and Galileo had regarded simply as a paradox was made by
Dedekind, in his clarification of the calculus, the basis of a definition
of infinite assemblages.
Although Bolzano's ideas indicate the direction in which the final
formulation of the calculus lay and which much of the thought of the
nineteenth century was to follow, they did not constitute the decisive
influence in determining this. His work remained largely unnoticed
until rediscovered by Hermann Hankel more than a half century
later. 13 Fortunately, however, the mathematician A. L. Cauchy pur-
sued similar ideas at about the same time and was successful in
establishing these as basic in the calculus.
Cauchy rivaled Euler in mathematical productivity, contributing
some 800 books and articles on almost all branches of the subject. 14
Among his greatest contributions are the rigorous methods which he
introduced into the calculus in his three great treatises: the Cours
d 'analyse de I'Ecole Poly technique (1821), Resume des leqons sur le
calcul infinitesimal (1823), and Leqons sur le calcul differentiel (1829). 15
Through these works Cauchy did more than anyone else to impress
upon the subject the character which it bears at the present time.
We have seen the notion of a limit develop gradually out of the Greek
method of exhaustion, until it was expressed by Newton in the
Principia. It was more definitely invoked by Robins, D'Alembert,
and L'Huilier as the basic concept of the calculus, and as such was
included by Lacroix in his textbooks. Throughout this long period,
however, the limit concept lacked precision of formulation. This
resulted from the fact that it was based on geometrical intuition. It
could hardly have been otherwise, inasmuch as during this time the
ideas of arithmetic and algebra were largely established upon those
of geometrical magnitude. The calculus was interpreted by its in-
ventors as an instrument for dealing with relationships between
13 See Dickstein, "Zur Geschichte der Principien der Infinitesimalrechnung," p. 77; also
Hankel's article "Grenze," in Ersch und Gruber's Allgemeine Encyklopddie.
14 See Valson, La Vie et les travaux du Baron Cauchy, reviewed by Boncompagni, p. 57.
16 The first is found in (Euvres (2), III; the last two in (Euvres (2), IV.
272 The Rigorous Formulation
quantities involved in geometrical problems, and as such was largely
accepted by their successors. Euler and Lagrange, in a sense repre-
sented exceptions to this rule, for they wished to establish the calculus
on the formalism of their analytic function concept. However, even
they rejected the limit idea. Furthermore they unwittingly deferred
to the preconceptions of geometrical intuition when they uncritically
inferred that their methods were applicable to all continuous curves.
Although D'Alembert, L'Huilier, and Lacroix had prepared the
ground for Cauchy by popularizing the limit idea in their works, this
conception remained largely geometrical.
In the work of Cauchy, however, the limit concept became, as it
had in the thought of Bolzano, clearly and definitely arithmetical
rather than geometrical. Formerly, when illustrations of the notion
were desired, the one most likely to be called to mind was that of a
circle defined as the limit of a polygon. Such an illustration immedi-
ately served to bring up questions as to the manner in which this is
to be interpreted. Is it the approach to coincidence of the sides of the
polygon with the points representing the circle? Does the polygon
ever become the circle? Are the properties of the polygon and the
circle the same? It was questions such as these that retarded the
acceptance of the limit idea, for they were similar to those of Zeno
in demanding some sort of visualization of the passage from the one
to the other by which the properties of the first figure merge into
those of the second.
Quite recently the limit concept has been loosely interpreted in
the assertion that whether one calls the circle the limit of a polygon
as the sides are indefinitely decreased, or whether one looks upon it
as a polygon with an infinite number of infinitesimal sides, is im-
material, inasmuch as in either case in the end "the specific differ-
ence" between the polygon and the circle is destroyed. 16 Such an
appeal to geometrical intuition is quite irrelevant in the case of the
limit concept. In giving his definition, in his Cours d* analyse, Cauchy
divorced the idea from all reference to geometrical figures or magni-
tudes, saying: "When the successive values attributed to a variable
approach indefinitely a fixed value so as to end by differing from it
by as little as one wishes, this last is called the limit of all the others." 17
» Vivanti, // concetto d'infinitesimo, p. 39. " See (Euvres (2), III, 19; cf. also IV, 13.
The Rigorous Formulation 273
This is the most clear-cut definition of the concept which had been
given up to that time, although later mathematicians were to voice
objections to this also and to seek to make it still more formal and
precise. Cauchy's definition appealed to the notions of number, vari-
able, and function, rather than to intuitions of geometry and dy-
namics. Consequently, in illustrating the limit concept he said that an
irrational number is the limit of the various rational fractions which
furnish more and more approximate values of it. 18
Upon the basis of this arithmetical definition of limit, Cauchy then
proceeded to define that elusive term, infinitesimal. Ever since the
time that Greek mathematical speculation had hit upon the infinitely
small, this notion had been bound up with geometrical intuition of
spatial properties and had been regarded as a more or less fixed
minimum of extension. The concept had not thrived in arithmetic,
largely because unity was considered the numerical minimum, the
rational fractions having been treated as ratios of two numbers.
However, the seventeenth century saw the rapid rise of algebraic
methods in geometry, so that from the time of Fermat the infinitesimal
had been the concern of both algebra and geometry. Newton had
insisted that his method did not involve the consideration of minima
sensibilia, but his interpretation of the procedure in terms of an
ultimate ratio, rather than of a limiting number, made his evanescent
magnitudes appear such. Liebniz had been less definite and con-
sistent in denying the existence of actual infinitesimals, for he some-
times considered them as assignable, sometimes as inassignable, and
occasionally as qualitatively zero. However, the development during
the eighteenth century of the function concept, with its emphasis on
the relations between variables, led Cauchy to make the infinitesimal
nothing more than a variable. "One says that a variable quantity
becomes infinitely small when its numerical value decreases indefi-
nitely in such a way as to converge toward the limit zero." 19 An
infinitesimal was consequently not different from other variables,
except in the understanding that it is to take on values converging
toward zero as a limit.
In order to make the concept of the infinitesimal more useful and
to take advantage of the operational facility afforded by the Leibnizian
» (Euvres (2), III, 19, 341. » (Euvres (2), III, 19; IV, 16.
274 The Rigorous Formulation
views, Cauchy added definitions of infinitesimals of higher order.
Newton had restricted himself to infinitesimals, or evanescent quan-
tities, of the first order, but Leibniz had attempted to define those of
higher orders also. One of the second order, for example, he defined
as being to one of first order as the latter is to a given finite constant,
such as unity. Such a vague definition could not be consistently
applied, but D'Alembert had sought to correct it by an interpretation
in terms of limits. He had in a sense recognized that the infinitesimals
were to be regarded as variables and that orders of infinitely small
magnitudes were to be defined in terms of ratios of these; but his
work, as we have seen, lacked the precision of statement necessary
for its general acceptance. It thus remained for Cauchy to express the
idea of D'Alembert in the precise symbolism of the limit concept.
Cauchy defined an infinitesimal y = f(x) to be of order n with
lim/ y\ _ no lim/ y '
respect to an infinitesimal x if *-»o( -^zr e ) = and x _>ol -^+- € I = ± oo
€ having its classical significance — a positive constant, however small. 20
Here again one recognizes in Cauchy 's work the dominance of the
ideas of variable, function, and limit. Newton, Leibniz, and D'Alem-
bert had not distinguished clearly between independent and depend-
ent infinitesimals, but Cauchy incorporated this in his definition
when he spoke of the order of the infinitesimal, y, with respect
to another, x. Thus the latter is the independent variable which
may be given any sequence of values tending toward as a limit,
the corresponding sequence of values of y being determined from
the functional relationship between y and x. Then the limits of the
sequences of values of the ratios -^- 6 and -^+~ e are found, and the
order of the infinitesimal y thus determined. This is substantially
the same as the definition commonly given in present-day textbooks —
that y is said to be an infinitesimal of order n with respect to another
infinitesimal x if nm —„ is a constant different from zero.
In a similar manner Cauchy made rigorously clear the views on
orders of infinity which D'Alembert had expressed. Whereas Bolzano,
»Seeffi»w«(2),IV,281.
The Rigorous Formulation 275
thinking in terms of aggregation, had asserted the possibility of an
actual infinity, Cauchy, emphasizing variability, denied the possi-
bility of this because of the paradoxes to which such an assumption
appeared to lead. 21 He admitted only the potential infinite of Aris-
totle, and with D'Alembert interpreted the infinite as meaning simply
indefinitely large — a variable, the successive values of which increase
beyond any given number. 22 Orders of infinity he then defined in a
manner exactly analogous to that given for infinitesimals.
Having established the notions of limit, infinitesimal, and infinity,
Cauchy was able to define the central concept of the calculus — that
of the derivative. His formulation was precisely that given by Bolzano:
Let the function be y = f(x); to the variable x give an increment
/\y fix ~\~ t) — f\X)
Ax = i\ and form the ratio • — = . The limit of this
Ax i
ratio ("when it exists") as i approaches zero he represented by f(x),
and he called this the derivative of y with respect to xP This is, of
course, the differential quotient of L'Huilier, clarified by the applica-
tion of the function concept of Euler and Lagrange. It is made the
central concept of the differential calculus, and the expression "dif-
ferential" is then defined in terms of the derivative. The differential
thus represents simply a convenient auxiliary notion permitting the
application of the suggestive notation of Leibniz without the con-
fusion between increments and differentials which this symbolism had
engendered. Leibniz had considered differentials as the fundamental
concepts, the differential quotient being defined in terms of these; but
Cauchy reversed this relationship. Having defined the derivative in
terms of limits, he then expressed the differential in terms of the
derivative. If dx is a finite constant quantity h, then the differential
dy of y = f{x) is defined as f{x)dx. In other words, the differentials
(dy) . .,
dy and dx are quantities so chosen that their ratio - — - coincides
{dx)
with that of the "derniere raison" or the limit y' = f(x) of the ratio
Ay
— -, 24 This is practically the view of D'Alembert and L'Huilier, and
Ax
21 See Enriques, Historic Development of Logic, p. 135.
22 (Euvres (2), HI, 19 ff.; cf. also IV, 16.
■ (Euvres (2), IV, 22; cf. also pp. 287-89. M (Euvres (2) IV, 27-28; 287-89.
276 The Rigorous Formulation
even Leibniz had in a sense anticipated it when he said in 1684 that
dy was to dx as the ratio of the ordinate to the subtangent. To make
his work logical, however, Leibniz would have had to define the term
"subtangent" in terms of limits, thus making the limiting ratio the
derivative.
Cauchy, however, gave to the derivative and the differential a
formal precision which had been lacking in the definitions of his
predecessors. He was therefore able to give satisfactory definitions of
differentials of higher order also. The differential dy = f(x)dx is, of
course, a function of x and dx. Regarding dx as fixed, the function
f(x)dx will in turn have a derivative f'{x)dx and a differential d 2 y =
f'(x)dx. 2 In general d n y = f{x)dx n . Cauchy added that because the
nth derivative is the coefficient by which dx n is to be multiplied to
give d"y, this derivative is called the differential coefficient. 25
This statement is not to be understood as indicating that deriva-
tives of higher order are to be defined in terms of differentials of
higher order. The reverse is of course the case. Differentials have no
logical significance independent of that of derivatives. They were
retained by Cauchy simply as an auxiliary notion offering greater
operational facility than that afforded by derivatives. This fact has
led the mathematician Hadamard, in connection with a discussion on
the subject given in the Mathematical Gazette for the years 1934-36,
to disparage as meaningless the use of differentials of higher order in
expositions of the calculus. 26
Cauchy's definitions of the derivative and the differential are not
in any real sense new. They indicate rather a clarification, by the
application of the concepts of function, variable, and limit of a vari-
able, of others previously given. During the eighteenth century the
word function had generally designated an expression which could be
written down simply, in terms of variables and symbols of operations
commonly employed at the time. The nondifferentiable function of
Bolzano would not, of course, have been included under such an
understanding. However, early in the next century the work of J. B.
J. Fourier showed that quite arbitrary discontinuous curves could be
represented analytically by means of infinite series of trignometric
25 ffi«w«(2),IV, 301 ff.
26 Hadamard, "La Notion de diffSrentiel dans Penseignement."
The Rigorous Formulation 277
functions. As a consequence, the attitude toward the function concept
became broader. 27
The formalistic view that a function was a simple analytic ex-
pression gave way to the understanding that it was any relationship
between variables. With this attitude came also the recognition that
continuity in a curve did not depend upon its being expressible by
means of a single equation in continuous functions. This, in turn, led
to the realization that a new definition of continuity was necessary.
Cauchy's answer to the need was similar to the unnoticed one of
Bolzano: the function /(x) is continuous within given limits if between
these limits an infinitely small increment i in the variable x produces
always an infinitely small increment, f(x + i) — /(x), in the function
itself. 28 The expressions infinitely small are here to be understood, as
elsewhere in Cauchy's work, in terms of the indefinitely small and
limits: i. e., f(x) is continuous within an interval if the limit of the
variable /(x) as x approaches a is f(a), for any value of a within this
interval. In this definition the view of the preceding centuries is
reversed. Newton (implicitly) and Leibniz (explicitly) based the
validity of the calculus on the assumption, which Greek thought had
avoided, that, by a vague sense of continuity, limiting states would
obey the same laws as the variables approaching them. Cauchy made
the notion of continuity precisely mathematical and showed that this
depends upon the limiting idea and not vice versa. Furthermore, its
essence does not lie in a vague blending or unity or contiguity of
parts, as intuition seems to imply and as Aristotle had stated, but in
certain formal particulate arithmetical relationships, elaborated later
in the theory of sets of points, which in turn led to the definition of
the continuum.
With the new notion of continuity came a group of new problems.
It may have been remarked that throughout the period of indecision —
from Newton and Leibniz to Lagrange and Lacroix — the discussion
centered about the concepts of the differential and the derivative, to
the exclusion of that of the integral. The explanation of this is easily
found. From the time of the Greeks down to that of Pascal, areas
27 See Jburdain's two articles, "Note on Fourier's Influence on the Conceptions of
Mathematics," and "The Origin of Cauchy's Conceptions of a Definite Integral and of
the Continuity of a Function/'
28 (Euvres (2), III, 43; IV, 19-20, 278.
278 The Rigorous Formulation
had been found by various devices equivalent to summations of
elements. When properly interpreted in terms of the limit concept,
these methods represented the counterpart of what is now called the
definite integral. With Barrow, Newton, and Leibniz, however, the
remarkable discovery was made that the problem of finding areas
was simply the inverse of that of determining tangents to curves.
Inasmuch as convenient algorithms — those of fluxions and of dif-
ferentials — were developed in connection with the latter class of prob-
lems, by the mere process of inversion the determinations of quad-
ratures could be systematized.
The inverse of the fluxion Newton called the fluent. Leibniz had
himself defined the integral as a sum of differentials, although he
recognized it also as the inverse of the differential and had deter-
mined it in accordance with this fact. These inverses of the fluxion and
the differential are the equivalents of what is now called the anti-
derivative, or the primitive, or the indefinite integral, or sometimes
simply the integral. During the period we have been discussing, this
aspect of the integral as an inverse prevailed over that of the integral
as a sum. John Bernoulli, in the formal development of the calculus
summatorius of Leibniz, gave up the definition of the integral as a sum
and called it definitely the inverse of the differential. He conceived the
object of the integral calculus as that of finding, from a given relation
among the differentials, the relation of the quantities themselves. 29
Euler used the sum conception to find the approximate values of
definite integrals, but because he interpreted the differential as zero,
he rejected the Leibnizian view of integration as a process of summa-
tion and followed John Bernoulli in defining the integral as the inverse
of the differential. 30 L'Huilier went so far in emphasizing the integral
as the inverse of the derivative that he suggested substituting the
expression "rapport integral" for "somme integrate." 31 Lagrange like-
wise considered the problem of the integral calculus as that of deter-
mining from the "fonctions derivees" the original function; 32 and
Lacroix said that the object of the integral calculus was to determine
from the differential coefficients the functions from which they were
29 Die erste Integralrechnung, p. 3; cf. also p. 8; also Opera omnia, III, 387.
80 Opera omnia, XI, 7.
31 Exposition Slementaire, p. 32. Cf. p. 144. 82 CEuvres, III, 443.
The Rigorous Formulation 279
derived. 53 Bolzano similarly defined the integral as the inverse of the
derivative. 34
The result of this tendency in the calculus was that logically the
definition of the integral during this time rested immediately upon
that of the differential, and the latter became the basis of discussions
on the validity of the operations and conceptions of the calculus.
Developments during the early nineteenth century, however, intro-
duced new points of view. These led to the reinstatement by Cauchy
of the notion of the (definite) integral as a limit of a sum, and made .
necessary two independent definitions of the two fundamental con-
cepts of the calculus, those of the derivative and the integral.
,. f(x _f_ fj\ — f(x)
Inasmuch as the derivative has been defined as , im ,
*->o h
we see from the definition of continuity given by Bolzano and Cauchy
that the existence of this implied the continuity of the function at the
value in question, although the converse is not true. The existence of
an integral in the eighteenth-century sense, that is, as an anti-
derivative, is therefore bound up with the question of continuity.
However, even discontinuous curves apparently have an area, and so
discontinuous functions may allow of an integral in the Leibnizian
sense. Cauchy therefore restored the character of the definite integral
as a sum. For a function y = f(x), continuous in the interval from
Xo to X, he formed the characteristic sum of the products S H =
(xi - x )f(xo) + (#2 - xi)f(xi) + . . . + (X — x H _ i )f(x n _ l ). If the
absolute values of the differences x i + l — x { decrease indefinitely, the
value of S n will "finally attain a certain limit" 5 which will depend
uniquely on the form of the function fix) and on the limiting values
x and X . . . "This limit is called a definite integral." 35
Cauchy cautioned that the symbol of integration J" employed to
designate this limit was not to be interpreted as a sum, but rather
as a limit of a sum of this type. 36 Cauchy then brought out the fact
that although the two operations are defined independently of each
other, integration in this sense is the inverse of the process of differ-
« TraiiS du c ale id, n, 1-2. » Bolzano's Schriften, pp. 83-84.
K (Euvres (2), IV, 125; cf. also Jourdain, "The Origin of Cauchy's Conceptions," pp.
664 ft.; "The Theory of Functions with Cauchy and Gauss," p. 193.
»«<E»w«(2), IV, 126.
280 The Rigorous Formulation
entiation. He showed that if f(x) is a continuous function, the function
defined as the definite integral F(x) = jl f(x)dx has as its derivative
the function f(x). 37 This was perhaps the first rigorous demon-
stration of the proposition known as the fundamental theorem of the
calculus. 38
Cauchy's definition of the definite integral allows of extension, with
slight modifications, to functions which have discontinuities within
the interval of definition. If, for example, the function f{x) is dis-
continuous at the point X within the interval x to X, the definite
integral from x to X is defined as the limit, if it exists, of the sum
Ixo ~ e f(%)dx + Jxo + e f( x )dx, as c becomes indefinitely small. 39
Discontinuous functions have come to play a significant role in
mathematics and science, and the view of the integral as a sum has
been that upon which the theory of integration has largely developed
since the time of Cauchy. From this view, for example, the integral
of Lebesgue has developed. 40 The manipulation of infinite series, such
as those entering in the definition of the definite integral, had gone
on for well over a century before the time of Cauchy, but the need
for considering the convergence of these had not been strongly felt
until about the opening of the nineteenth century. The term "conver-
gent series" seems first to have been used, although in a somewhat
restricted sense, a century and a half earlier by James Gregory; 41
but the general lack of rigor in the work of the eighteenth century
was uncongenial to the precision of thought necessary to develop this
idea. Varignon, at the beginning of the century, and Lagrange, at its
close, had gone a step in this direction by saying that no series could
safely be used unless one investigated the remainder. 42
Nevertheless, no general definition, or theory, of convergent infinite
series had been given, and Euler and Lacroix continued to employ
divergent series in their work. With the turn of the next century,
however, Abel, Bolzano, Cauchy, and Gauss all pointed out the
need for definitions and tests of convergence of infinite series before
the latter could legitimately be employed in mathematics. In this
* Ibid., pp. 151-52. M Saks, Tktorie de Vintigrale, pp. 122-23.
39 See Cauchy, (Euvres (2), I, 335 and (Euvres (1), I, 335.
40 Saks, Thiorie de I'intigrale, p. 125.
41 See Vera circuli et hyperbolae quadratura, p. 10.
42 Reiff, Geschichte der unendlichen Reihen, pp. 69-70, 155.
The Rigorous Formulation 281
respect the work of Cauchy, in particular, laid the foundation of the
theory of convergence and divergence through the wide influence which
his work exerted upon his contemporaries. Cauchy defined a series as
convergent if, for increasing values of n, the sum S n approaches indefi- \
nitely a certain limit S, the limit S in this case being called the sum
of the series. 43 Cauchy here showed clearly that the limit notion is
involved, as it was also in differentiation and integration and in
defining continuity. Furthermore, he pointed out that it is only in i
this sense that an infinite series may be regarded as having a sum. ;
In other words, Zeno's paradox of the Achilles is to be answered in
precisely such ideas, based upon the limit concept.
Cauchy went on in this work to try to prove what has become
known as Cauchy's theorem — that a necessary and sufficient condi-
tion that the sequence converge to a limit is that the difference be-
tween S P and S q for any values of p and q greater than n can be
made less in absolute value than any assignable quantity by taking
n sufficiently large. A sequence satisfying this condition is now said
to converge within itself. The necessity of the condition follows im-
mediately from the definition of convergence, but the proof of the
sufficiency of the condition requires a previous definition of the system
of real numbers, of which the supposed limit 5 is one. Without a
definition of irrational numbers, this part of the proof is logically im-
possible.
Cauchy had stated in his Cours d 'analyse that irrational numbers
are to be regarded as the limits of sequences of rational numbers.
Since a limit is defined as a number to which the terms of the sequence
approach in such a way that ultimately the difference between this
number and the terms of the sequence can be made less than any
given number, the existence of the irrational number depends, in the
definition of limit, upon the known existence, and hence the prior
definition, of the very quantity whose definition is being attempted.
That is, one cannot define the number V 2 as the limit of the sequence
1, 1.4, 1.41, 1.414, . . . because to prove that this sequence has a limit
one must assume, in view of the definitions of limit and convergence,
the existence of this number as previously demonstrated or defined.
Cauchy appears not to have noticed the circularity of the reasoning
« (Euvres (2), HI, 114.
282 The Rigorous Formulation
in this connection, 44 but tacitly assumed that every sequence con-
verging within itself has a limit. That is, he felt that the existence of a
number possessing the external relationship expressed in the definition
of convergence and the sum of the series, would follow from the inner
relations expressed in the Cauchy theorem. This idea may have been
based upon the very thing that he and Bolzano had sought to avoid —
that is, upon preconceptions taken over from geometry. The attempt
to base the idea of number upon that of the geometrical line had given
rise to the Pythagorean difficulty of the incommensurable and the
ensuing development of the calculus. It had likewise suggested to
Gregory of St. Vincent, two centuries before Cauchy, that the sum
of an infinite geometrical progression could be represented by the
length of a line segment and that the series could therefore be thought
of as having a limit. However, in order to make the limit concept of
analysis independent of geometry, mathematicians of the second half
of the nineteenth century attempted to frame definitions of irrational
number which did not make use of the definition of a limit.
The geometrical intuitions which intruded themselves into Cauchy's
view of irrational number likewise led him erroneously to believe that
the continuity of a function was sufficient for its geometrical repre-
sentation and for the existence of a derivative. 46 A. M. Ampere also
had been led by geometric preconceptions similar to those of Cauchy
to try to demonstrate the false proposition that every continuous
function has a derivative, except for certain isolated values in the
interval. 46 Bolzano had in his manuscripts of about this time given an
example showing the falsity of such an opinion, but it remained for
Weierstrass to make this fact known.
With Cauchy, it may safely be said, the fundamental concepts of
the calculus received a rigorous formulation. Cauchy has for this
reason commonly been regarded as the founder of the exact differential
calculus in the modern sense. 47 Upon a precise definition of the notion
of limit, he built the theory of continuity and infinite series, of the
derivative, the differential, and the integral. Through the popularity
of his lectures and textbooks, his exposition of the calculus became
44 Cf. Pringsheim, "Nombres irrationnels et notion de limite," p. 180.
45 Cf. Jourdain, "The Theory of Functions with Cauchy and Gauss.''
48 See Pringsheim, "Principes fondamentaux de la theorie des fonctions."
47 Klein, Elementary Mathematics from an Advanced Standpoint, p. 213.
The Rigorous Formulation 283
that generally adopted and the one which has been accepted down to
the present time. Nevertheless, the use of the infinitely small per-
sisted for some time. S. D. Poisson, in his Traite de mecanique which
appeared in several editions in the first half of the nineteenth century
and which was long a standard work, used exclusively the method of
infinitesimals. These magnitudes, "less than any given magnitude of
the same nature," he held to have a real existence. They were not
simply "a means of investigation imagined by geometers." 48 The
object of the differential calculus he consequently regarded as the
determination of the ratio of infinitely small quantities, in which
infinitesimals of higher order were neglected; 49 and the integral was
the inverse of the differential quotient.
A. A. Cournot likewise opposed the work of Cauchy, although upon
somewhat different grounds. In his Traite elementaire de la theorie des
jonctions et du calcul infinitesimal of 1841, he asserted that his taste
for the philosophy of science prepared him to treat the metaphysics
of the calculus. 50 In presenting this, his attitude resembled some-
what that of Carnot. He held that the theories of Newton and Leibniz
complemented each other, and that the method of Lagrange repre-
sented simply a return to the views of Newton. 51 The infinitely small
"existed in nature" as a mode of generation of magnitudes according
to the law of continuity, although it could be defined only indirectly
in terms of limits. 52
However, Cournot protested that concepts exist in the under-
standing, independently of the definition which one gives to them.
Simple ideas sometimes have complicated definitions, or even none.
For this reason he felt that one should not subordinate the precision
of such ideas as those of speed or the infinitely small to logical defi-
nition. 53 This point of view is diametrically opposed to that which
has dominated the mathematics of the last century. The tendency in
analysis since the time of Cournot has been toward ever-greater care
in the formal logical elaboration of the subject. This trend, initiated
in the first half of the nineteenth century and fostered largely by
Cauchy, was in the second half of that century continued with notable
success by Weierstrass.
* Traits de mfcanique, I, 13-14. * Ibid., pp. 14-16.
60 Traite elementaire, Preface. 61 Ibid. B2 Ibid., pp. 85-88. M Ibid., p. 72.
284 The Rigorous Formulation
In spite of the care with which Cauchy worked, there were a num-
ber of phrases in his exposition which required further explanation.
The expressions "approach indefinitely," "as little as one wishes,"
"last ratios of infinitely small increments," were to be understood in
terms of the method of limits, but they suggested difficulties which
had been raised in the preceding century. The very idea of a variable
approaching a limit called forth vague intuitions of motion and the
generation of quantities. Furthermore, there were, in Cauchy's pres-
entation, certain subtle logical gaps. One of these was the failure to
make clear the notion of an infinite aggregate, which is basic in his
work in infinite sequences, upon which the derivative and the integral
are built. Another lacuna is evident in his omission of a clear defi-
nition of that most fundamental of all notions — number — which is
absolutely essential to the definition of limits, and therefore to that
of the concepts of the calculus. The first of these points had been
touched upon by Bolzano, but the theory was not further developed
until much later, largely through the efforts of Georg Cantor. In the
second matter the difficulty is essentially that of a vicious circle in
the definition of irrational numbers, and this Weierstrass sought to
resolve.
Although it was Cauchy who gave to the concepts of the calculus
their present general form, based upon the limit concept, the last
word on rigor had not been said, for it was Karl Weierstrass 54 who
constructed a purely formal arithmetic basis for analysis, quite
independent of all geometric intuition. Weierstrass in 1872 read a
paper in which he showed what had been known to Bolzano sometime
before — that a function which is continuous throughout an interval
need not have a derivative at any point in this interval. 55 Previously
it had been generally held, upon the basis of physical experience, that
a continuous curve necessarily possessed a tangent, except perhaps at
certain isolated points. From this it would follow that the correspond-
ing function should in general possess a derivative. Weierstrass, how-
ever, demonstrated conclusively the incorrectness of such suggestions
w See Poincare, "L'CEuvre mathematique de Weierstrass"; cf. also Pierpont, "Mathe-
matical Rigor," pp. 34r-36.
65 This seems to have been presented by Weierstrass in his lectures as early as 1861.
See Pringsheim, "Principes fondamentaux," p. 45, n.: Voss, "Calcul differentiel," pp.
260-61. ™
The Rigorous Formulation 285
of experience. This he did by forming the nondifferentiable contin-
OS
uous function f(x) = 2 b n cos{a n Tx), where x is a real variable, a an
» = o
odd integer, and b a positive constant less than unity such that
ab > 1 + — .«*
2
Since that time many other such functions have become known, and
we may even say that, in spite of geometric intuition, of all contin-
uous functions those with tangents at some points are the exceptions."
Intuition has been even more discredited as a guide by the fact that
we can have continuous curves defined by motion, which yet have
no tangents. 58
Inasmuch as it was apparent to Weierstrass that intuition could not
be trusted, he sought to make the bases of his analysis as rigorously
and precisely formal as possible. He did not present his work on the
elements of the calculus in a number of treatises, as had Cauchy, nor
even in a series of papers. His views became known, rather, through
the work of students who attended his lectures. 69
In order to secure logical exactitude, Weierstrass wished to estab-
lish the calculus (and the theory of functions) upon the concept of
number alone, 60 thus separating it completely from geometry. To do
this it was necessary to give a definition of irrational number which
should be independent of the limit idea, since the latter presupposes
the former. Weierstrass was thus led to make profound investigations
into the principles of arithmetic, particularly with respect to the theory
of irrationals. In this work Weierstrass did not go into the nature
of the whole number itself, but began with the concept of whole
number as an aggregate of units enjoying one characteristic property
in common, whereas a complex number was to be thought of as an
aggregate of units of various species enjoying more than one char-
56 See Weierstrass, Mathematische Werke, II, 71-74; Mansion, "Fonction continue sans
derivee de Weierstrass."
67 Cf. Voss, "Calcul differentiel," pp. 261-62.
M Neikirk, "A Class of Continuous Curves Defined by Motion Which Have No Tan-
gents Lines."
69 See, for example, Pincherle, "Saggio di una introduzione alia teoria delle funzioni
analitiche secondo i principii del Prof. C. Weierstrass." Cf. Merz, A History of European
Thought in the Nineteenth Century, II, 703.
60 Jourdain, "The Development of the Theory of Transfinite Numbers," 1908-9, n., p.
298; cf. also p. 303.
286 The Rigorous Formulation
acteristic property. All rational numbers can then be denned by
introducing convenient classes of complex numbers. Thus the number
3f is made up of 3 a and 2/3, where a is the principle unit and /3 is an
aliquot part, §, taken as another element. A number is then said to.
be determined when we know of what elements (of which there is an
infinite number) it is composed and the number of times each occurs.
In this theory the number V 2 is not denned as the limit of the sequence
1, 1.4, 1.41, ... , nor is the idea of sequence brought in; it is simply
the aggregate itself in any order l a , 40, I7, . . . where a is the prin-
ciple unit and p, y, . . . are certain of its aliquot parts, and where the
aggregate is, of course, subject to the condition that the sum of any
finite number of elements is always less than a certain rational num-
ber. We can now prove, if we wish, that this number is the limit of
the variable sequence l a ; la, 40; U, 4/3, 1 T ; . . . , thus correcting the
logical error arising in Cauchy's theory of number and limits. 61 In a
sense, Weierstrass settles the question of the existence of a limit of a
convergent sequence by making the sequence (really he considers an
unordered aggregate) itself the number or limit.
In making the basis of the calculus more rigorously formal, Weier-
strass also attacked the appeal to intuition of continuous motion which
is implied in Cauchy's expression — that a variable approaches a limit.
Previous writers generally had denned a variable as a quantity or
magnitude which is not constant; but since the time of Weierstrass it
has been recognized that the ideas of variable and limit are not
essentially phoronomic, but involve purely static considerations.
Weierstrass interpreted a variable * as simply a letter designating any
one of a collection of numerical values. 62 A continuous variable was
likewise defined in terms of static considerations: If for any value
x of the set and for any sequence of positive numbers 5 i, 5 2 , . • • ,
8„, however small, there are in the intervals x — g,-, x + 8{ others of
the set, this is called continuous. 63
Similarly, for a continuous function Weierstrass gave a definition
equivalent to those of Bolzano and Cauchy, but having greater clarity
and precision. To say that f(x + A*) - f(x) becomes infinitesimal,
61 Ibid., 1908-9, pp. 303 ff. Cf. Russell, Principles of Mathematics, pp. 281 ff.; Pring-
sheim, "Nombres irrationnels et notion de limite," pp. 149 ff.; Pincherle, op. cit., pp. 179 ff.
« Pincherle, op. cit., p. 234. « Ibid., p. 236.
The Rigorous Formulation 287
or becomes and remains less than any given quantity, as A* ap-
proaches zero, calls to mind either the infinitely small or else vague
notions of mobility. Weierstrass defined f{x) as continuous, within
certain limits of x, if for any value x in this interval and for an arbi-
trarily small positive number e, it is possible to find an interval about
x such that for all values in this interval the difference f(x) — f(x )
is in absolute value less than e ; 64 or, as Heine was led by the lectures
of Weierstrass to express it, if, given any e , an v can be found such
that for V < \, the difference fix ± v) — f(x) is less in absolute
value than e. 65
The limit of a variable or function is similarly defined. The number
L is the limit of the function f(x) for x = x if, given any arbitrarily
small number c, another number 5 can be found such that for all
values of x differing from x by less than 5, the value of f(x) will
differ from that of L by less than e. 66 This expression of the limit idea,
in conjunction with Cauchy's definitions of the derivative and the
integral, supplied the fundamental conceptions of the calculus with a
precision which may be regarded as constituting their rigorous formula-
tion. There is in this definition no reference to infinitesimals, so that
the designation "the infinitesimal calculus," which is used even today,
is shown to be inappropriate. Although a number of mathematicians,
from the time of Newton and Leibniz to that of Bolzano and Cauchy,
had sought to avoid the use of infinitely small quantities, the un-
equivocal symbolism of Weierstrass may be regarded as effectively
banishing from the calculus the persistent notion of the fixed infin-
itesimal.
During the eighteenth century there had been a lively argument,
both in connection with the prime and ultimate ratio of Newton
and with the differential quotient of Leibniz, as to whether a variable
which approaches its limit can ever attain it. This is essentially the
crux of Zeno's argument in the Achilles. In the light of the precision
of the Weierstrassian theory of limits, however, the question is seen to
be entirely inapposite. The limit concept does not involve the idea of
approaching, but only a static state of affairs. The single question
M Ibid., p. 246.
65 Heine, "Die Elemente der Funktionenlehre," p. 182; cf . also p. 186.
M See Stolz, Vorlesungen fiber allgemeine Arithmetik, I, 156-57; cf. also Whitehead, An
Introduction to Mathematics, pp. 226-29.
288 The Rigorous Formulation
amounts really to two : first, does the variable f(x) have a limit L
for the value a of x. Secondly, is this limit L the value of the function
for the value a of x. If f(a) = L, then one can say that the limit
of the variable for the value of x in question is the value of the
variable for this value of x, but not that/(V) reaches j '(a) or L, for this
latter statement has no meaning.
In retrospect, it is pertinent to remark that whereas the idea of
variability had been banned from Greek mathematics because it led
to Zeno's paradoxes, it was precisely this concept which, revived in
the later Middle Ages and represented geometrically, led in the
seventeenth century to the calculus. Nevertheless, as the culmination
of almost two centuries of discussion as to the basis of the new anal-
ysis, the very aspect which had led to its rise was in a sense again
excluded from mathematics with the so-called "static" theory of the
variable which Weierstrass had developed. The variable does not
represent a progressive passage through all the values of an interval,
but the disjunctive assumption of any one of the values in the inter-
val. Our vague intuition of motion, although remarkably fruitful in
having suggested the investigations which produced the calculus, was
found, in the light of further elaboration in thought, to be quite
inadequate and misleading. What, then, about that obscure and
elusive feeling for continuity which colors so much of our thought?
Is that baseless also? What about the idea of infinity upon which
Bolzano had speculated and which Weierstrass had used, somewhat
covertly, in his definition of irrational number? Can this be given a
consistent definition? These questions were investigated largely by
Dedekind and Cantor, two mathematicians who were thinking along
lines similar to those which Weierstrass had followed in seeking a sat-
isfactory definition of irrational numbers.
The year 1872 was, for a number of reasons, a significant one in the
history of the foundations of the calculus. It saw, besides the pres-
entation by Weierstrass of his continuous nondifferentiable function
and the publication by one of his students of Weierstrass' lectures on
the elements of arithmetic, 67 the appearance of the following: Nouveau
precis d' analyse infinitesimale of Charles Meray; a paper in Crelle's
Journal by Eduard Heine on "Die Elemente der Funktionenlehre" ;
67 Kossak, Die Elemente der Arithmetik.
The Rigorous Formulation 289
the first paper by Georg Cantor on the principles of arithmetic, which
appeared in the Mathematische Annalen as "Uber die Ausdehnung
eines Satzes aus der Theorie der trigonometrischen Reihen"; and the
Stetigkeit und die Irrationalzahlen of Richard Dedekind. Incidentally
the work of each of these men touched upon one and the same prob-
lem — that of formulating a definition of irrational number which
should be independent of that of the limit concept. 68 The work of
Weierstrass in this connection has already been described. With
respect to publication, this had been anticipated by Meray, who in
1869, in an article entitled "Remarques sur la nature des quantites
definies par la condition de servir de limites a des variables donnees,"
sought to resolve the vicious circle in the definitions of limit and
irrational number given by Cauchy. Three years later Meray further
elaborated his views, in his Nouveau precis.
It will be recalled that Bolzano and Cauchy had attempted to
prove that a sequence which converges within itself — that is, one for
which, given any e, however small, an integer N can be found such
that for n > N and for any integral value of p greater than the integer
n, the inequality \S n + P - Sj, < e will hold— converges in the sense of
external relations, that is, that it has a limit S. Meray, in this con-
nection, cut the Gordian knot by rejecting Cauchy's definition of
convergence in terms of the limit S. He called an infinite series con-
vergent if it converged within itself, according to Cauchy's theorem.
In this case one need not demonstrate the existence of an undefined
number S, which may be regarded as the limit. The word number in
the strict sense Meray reserved for the integers and rational fractions;
the converging sequence of rational numbers, which Meray called a
convergent "variante," he regarded as determining a number in the
broad sense, rational or irrational. He was somewhat vague as to
whether or not the sequence is the number. 69 If so, as is implied in the
case of irrationals, his theory is equivalent to that of Weierstrass,
although somewhat less explicitly expressed.
Attempts, similar to those of Weierstrass and Meray, to avoid the
88 For a full account of this work, together with bibliographical references, see Jourdain,
"The Development of the Theory of Transfinite Numbers"; see also Pringsheim, "Nombres
irrationnels et notion de limite," pp. 144 ff.
69 See "Remarques sur la nature des quantites"; also Nouveau pricis, pp. xv, 1-7; cf.
also Jourdain, "The Development of the Theory of Transfinite Numbers,'' 1910, pp. 28 ff .
290 The Rigorous Formulation
petitio principii in Cauchy's reasoning on limits and irrational num-
bers were developed and published, also in 1872, by Cantor and by
Heine. Meray had avoided the logical difficulty by taking \S n + p —
S n \ < e as the definition of convergence, instead of the condition
\S — S n \ < e, which presupposes the demonstrated existence of S.
Likewise in Weierstrass' definition the irrational numbers are ex-
pressed in an analogous manner, not as limits but as entire infinite
groups of rational numbers. The work of these men led Heine and
Cantor to express similar views. Rather than postulate the existence
of a number S, which is the limit of an infinite series which converges
within itself, they considered S, not exactly as determined by the
series, as Meray had somewhat indecisively held, but as defined by
the series — as simply a symbol for the series itself. 70 Their definitions
resemble the view of Weierstrass, with the addition of the condition
of Meray that ^ (S n + P - S H ) = for p arbitrary. This condition is
equivalent to that of Weierstrass — whose aggregates were such that
however summed in finite number, the sum was to remain below a
certain limit — but is expressed in a rather more convenient form.
Still another attempt along these lines was made by Dedekind.
Weierstrass had been lecturing on the theory of functions in 1859
and had in this way been led to investigations concerning the founda-
tions of arithmetic. In like fashion Dedekind admitted that his atten-
tion was directed to these matters when in 1858 he found himself
obliged to lecture for the first time on the elements of the differential
calculus. 71 In discussing the notion of the approach of a variable
magnitude to a fixed limiting value, he had recourse, as had Cauchy
before him, to the evidence of the geometry of continuous magnitude.
He felt, however, that the theory of irrational numbers, which lay at
the root of the difficulty in the limit concept, should be developed
out of arithmetic alone, if it were to be rigorous. 72
Dedekind 's approach to the problem was somewhat different from
that of Weierstrass, Meray, Heine, and Cantor in that, instead of
70 Heine, "Die Elemente der Funktionenlehre," pp. 174 ff.; Cantor, Gesammelte Abltand-
lungen, pp. 92-102, 185-86. See also the articles by Jourdain, "The Development of the
Theory of Transfinite Numbers," 1910, pp. 21-43, and "The Introduction of Irrational
Numbers." Cf. Russell, The Principles of Mathematics, pp. 283-86.
71 Dedekind, Essays on the Tlieory of Numbers, p. 1.
» Ibid., pp. 1-3; cf. also p. 10.
The Rigorous Formulation 291
considering in what manner the irrationals are to be defined so as to
avoid the vicious circle of Cauchy, he asked himself what there is in
continuous geometrical magnitude which resolved the difficulty when
arithmetic apparently had failed: i. e., what is the nature of con-
tinuity? Plato had sought to find this in a vague flowing of magni-
tudes; Aristotle had felt that it lay in the fact that the extremities of
two successive parts were coincident. Galileo had suggested that it
was the result of an actually infinite subdivision — that the continuity
of a fluid was in this respect to be contrasted with the finite, dis-
continuous subdivision illustrated by a fine powder. The philosophy
and mathematics of Leibniz had led him to agree with Galileo that
continuity was a property concerning disjunctive aggregation, rather
than a unity or coincidence of parts. Leibniz had regarded a set as
forming a continuum if between any two elements there was always
another element of the set. 73
The scientist Ernst Mach likewise regarded this property of the
denseness of an assemblage as constituting its continuity, 74 but the
study of the real number system brought oat the inadequacy of this
condition. The rational numbers, for example, possess the property of
denseness and yet do not constitute a continuum. Dedekind, thinking
along these lines, found the essence of the continuity of a line to be
brought out, not by a vague hang-togetherness, but in the nature of
the division of the line by a point. He saw that in any division of the
points of a line into two classes such that every point of the one is to
the left of every point of the other, there is one and only one point
which produces this division. This is not true of the ordered system of
rational numbers. This, then, was why the points of a line formed a
continuum, but the rational numbers did not. As Dedekind expressed
it, "By this commonplace remark the secret of continuity is to be
revealed." 75
It is obvious, then, in what way the domain of rational numbers is
to be rendered complete to form a continuous domain. It is only neces-
sary to assume the Cantor-Dedekind axiom that the points of a line
can be put into one-to-one correspondence with the real numbers.
Arithmetically expressed, this means that for every division of the
73 Pkilosophische Schriften, II, 515.
74 Die Principien der Warmelehre, p. 71. 76 Essays on the Theory of Numbers, p. 11.
292 The Rigorous Formulation
rational numbers into two classes such that every number of the first,
A, is less than every number of the second, B, there is one and
only one real number producing this Schnitt, or "Dedekind Cut."
Thus if we divide the rational numbers into two classes A and B,
such that A contains all those whose squares are less than two and
B all those whose squares are more than two, there is, by this axiom
of continuity, a single real number — written in this case as V 2 — which
produces this division. Furthermore, this cut constitutes the definition
of the number V 2. Similarly, any real number is defined by such
a cut in the rational number system. This postulate makes the domain
of real numbers continuous, in the sense that the straight line has this
property. Moreover, the real number of Dedekind is in a sense a crea-
tion of the human mind, independent of intuitions of space and time.
The calculus had been generally recognized as dealing with contin-
uous magnitude, but before this time no one had explained precisely
the sense in which this was to be accepted. Symbols for variables
had displaced the idea of geometrical magnitude, but Cauchy im-
plied a geometrical interpretation of a continuous variable. Dedekind
showed that it was not, as had frequently been held, 76 the apparent
freedom from the discreteness of the rational numbers which made
geometrical quantities continuous, but only the fact that the points
in them formed a dense, perfect set. On completing the number system
in the manner suggested by this fact — that is, by adopting Dedekind's
postulate — this system was made continuous also. Now the funda-
mental theorems on limits could be proved rigorously 77 and without
recourse to geometry, as Dedekind pointed out, on the basis of his
new definition of real number. 78 Geometry having pointed the way
to a suitable definition of continuity, it was in the end excluded from
the formal arithmetical definition of this concept.
The Dedekind Cut is in a sense equivalent to the definitions of real
number given by Weierstrass, Meray, Heine, and Cantor. 79 Bertrand
76 See Drobisch, "Ueber den Begriff des Stetigen und seine Beziehungen zum Calcul,"
p. 170.
77 It should be noted, however, that Dedekind's definition of number has recently been
criticized as involving a vicious circle. See Weyl, "Der Circulus vitiosus in der heutigen
Begriindung der Analysis."
78 Essays on the Theory of Numbers, p. 27; cf. also pp. 35-36.
79 See J. Tannery, Review of Dantscher, Vorlesungen iiber die Weierstrassche Theorie der
irrationalen Zahlen.
The Rigorous Formulation 293
Russell followed the line of thought suggested by these men, in at-
tempting another formal definition of real number. He felt that the
definitions previously given either disregarded the question of the
existence of the irrational numbers or artificially postulated new
numbers, leaving some doubt as to just what they are. He suggested
that a real number be defined as a whole "segment" of the rational
numbers. The number V2, for example, is defined as the ordered
aggregate of all rational numbers whose squares are less than two.
That is, instead of postulating an element dividing the rational num-
bers into two classes, as Dedekind did, he would merely take one of
Dedekind's classes and make it, rather than the cutting element, the
number. 80 This obviates the necessity of introducing any conception
other than that of rational number and segment of rational numbers.
According to this view, there is no need to create the irrational num-
bers; they are at hand in the system of rational numbers, as they had
been also in the somewhat more involved doctrine of Weierstrass.
The object of all the above efforts in the establishment of the real
number was to give a formal logical definition which should be inde-
pendent of the implications of geometry and which should avoid the
logical error of defining irrational numbers in terms of limits, and
vice versa. From these definitions, then, the basic theorems on limits
in the calculus can be derived without circularity in reasoning. The
derivative and the integral are thus established directly on these
definitions, and are consequently divested of any character connected
with sensory perception, such as rate of change or surface area. Geo-
metrical conceptions cannot be made sufficiently explicit and precise,
as we have seen during our consideration of the long period of devel-
opment of the calculus. Thus the required rigor was found in the
application of the concept of number, made formal by divorcing it
from the idea of geometrical quantity. From the definitions of number
given above, we see that it is not magnitude which is basic, but order.
This is brought out most clearly in the definitions given by Dedekind
and Russell, these involving only ordered classes of elements. The
same is true, however, of the other systems, which have the dis-
advantage of requiring new definitions of equality before this can be
made clear. The essential characteristic of the number two is not its
80 Russell, Introduction to Mathematical Philosophy, p. 72.
294 The Rigorous Formulation
magnitude, but its place in the ordered aggregate of real numbers.
The derivative and the integral, although still defined as limits of
characteristic quotients and sums respectively, have, as a result, ulti-
mately become, through the definition of number and limit, not
quantitative but ordinal concepts. The calculus is not a branch of the
science of quantity, but of the logic of relations.
Dedekind's work not only met the need for a definition of number
independent of that of limit, but in addition gave an explanation of
the nature of continuous magnitude. Bolzano, Cauchy, and others
had given definitions of a continuous function of an independent
variable. A continuous, independent variable was tacitly understood
as one which could take on all values in an interval corresponding to
the points of a line segment. The arithmetization of 1872, however,
went beyond the geometrical picture and expressed formally, in terms
of ordered aggregates, what was meant by a continuous variable or
ensemble. The conditions were: first, that the values or elements
should form an ordered set; second, that this should be a dense set —
that is, between any two values or elements, there should always be
others; and third, that the set should be perfect — that is, if the ele-
ments are divided, as in a Dedekind Cut, there should always be one
which produces this cut.
This definition is far removed from any appeal to empiricism and
from the picture of a smooth, unbroken "oneness" or cohesiveness,
which instinctive feeling associates with the notion of continuity. It
specifies only an infinite, discrete multiplicity of elements, satisfying
certain conditions — that the set be ordered, dense, and perfect. This
is the sense in which one is to interpret the remark that the calculus
deals with continuous variables; the sense in which one is to interpret
Newton's phrase "prime and ultimate ratios," or the ultimate rela-
tionship between the differentials which Leibniz thought subsisted by
virtue of the law of continuity. The introduction of uniform motion
into Newton's method of fluxions was an irrelevant evasion of the
question of continuity, disguised by an appeal to intuition. There is
nothing dynamic in the idea of continuity, nor, so far as we know, is
the converse necessarily true. By sense perception we are apparently
unable to conclude whether or not we are dealing in motion with a
continuum. The experiments of Helmholtz, Mach, and others have
The Rigorous Formulation 295
shown that the physiological spaces of touch and sight are themselves
discontinuous. 81
The continuity of time which Barrow and Newton regarded as
assured by its relentless even flow is now seen to be simply a hy-
pothesis. Mathematics is unable to specify whether motion is con-
tinuous, for it deals merely with hypothetical relations and can make
its variable continuous or discontinuous at will. The paradoxes of
Zeno are consequences of the failure to appreciate this fact and of
the resulting lack of a precise specification of the problem. The dy-
namic intuition of motion is confused with the static concept of con-
tinuity. The former is a matter of scientific description a posteriori,
whereas the latter is a matter solely of mathematical definition a
priori. The former may consequently suggest that motion be defined
mathematically in terms of continuous variables, but cannot, because
of the limitations of sensory perception, prove that it must be so
defined. If the paradoxes of Zeno are thus stated in the precise mathe-
matical terminology of continuous variables and of the derived con-
cepts of limit, derivative, and integral, the seeming contradictions
resolve themselves. The dichotomy and the Achilles depend upon the
question as to whether or not the sets involved are perfect. 82 The
stade is answered upon the basis of dense sets, and the arrow by the
definition of instantaneous velocity, or the derivative.
The mathematical theory of continuity is based, not on intuition,
but on the logically developed theories of number and sets of points.
The latter, however, depend, in turn, upon the idea of an infinite
aggregate, an idea which Zeno had invoked to fortify his arguments.
Zeno's appeal to the infinite was based upon the supposed incon-
ceivability of the notion of completing in a finite time an infinite
number of steps. It is again the scientific description a posteriori
which he questioned, but, so far as we know, there is no way of proving
or disproving the possibility, not only of the existence of infinite
aggregates in the physical sense, but also of the execution in thought
of an infinite number of steps in connection with aggregates, whether
finite or infinite. Since science cannot answer this point, the question
may become a hypothetical mathematical one.
81 Enriques, Problems of Science, pp. 211-12.
82 Cf. Helmholtz, Counting and Measuring, p. xviii; Cajori, "History of Zeno's Argu-
ments on Motion," p. 218.
296 The Rigorous Formulation
Mathematics, moreover, requires a theory of the infinite in its
definitions of number and continuity. The question of the mathe-
matical existence, i. e., of the consistent logical definition, of infinite
aggregates therefore remained to be answered. Galileo had suggested
vaguely, and Bolzano had seen more clearly, that infinite ensembles
must have the paradoxical property that a part can be put into one-
to-one correspondence with the whole. This fact led Cauchy to deny
their existence; and indeed upon the basis of the work of Cauchy and
Weierstrass, one could have said that the infinite indicated nothing
more than the potentiality of Aristotle — an incompleteness of the
process in question. 83 Their infinitesimals were variables having zero
as their limit, and the limit concept involved only the definition of
number. However, this very definition of number implicitly pre-
supposes the prior existence of infinite aggregates, so that this ques-
tion could not indefinitely be avoided. Under the influence of Weier-
strass' work in the foundations of arithmetic, Dedekind and Cantor
sought a basis for the theory of infinite aggregates, 84 in order to com-
plete this work. This they found in Bolzano's paradox. Instead of
looking upon it as merely a strange property of infinite aggregates,
they made it the definition of an infinite set. Dedekind said "A system
5 is said to be infinite when it is similar to a proper part of itself; in
the contrary case 5 is said to be a finite system." 85 Under this defi-
nition, infinite aggregates exist as logically self-consistent entities, and
the definitions of real number are completed.
Cantor, with whom Dedekind corresponded in this connection, 86
was not satisfied with merely defining infinite sets. He wished to de-
velop the subject further. In a series of papers he reviewed the history
of the infinite from the time of Democritus to that of Dedekind, and
elaborated his theory of infinite ensembles, or Mengenlehre. Cantor's
doctrine of the mathematical infinite, which has been hyperbolized as
"the only genuine mathematics since the Greeks," 87 did not refer to
the potential infinity of Aristotle nor the syncategorematic infinity of
the Scholastics. These were bound up always with variability. 88
Cantor referred instead to the categorematic infinity of medieval
83 See Baumann, "Dedekind und Bolzano." M See Hilbert, "Uber das Unendliche."
85 Dedekind, Essays on the Theory of Numbers, p. 63.
86 See Georg Cantor, Gesammelte Abhandlungen.
87 See Bell, The Queen of the Sciences, p. 104. w Ibid., p. 180.
The Rigorous Formulation 297
philosophy — the eigenilich-Unendlichen. He felt, with some justice,
that Scholastic thought had handled this subject more as a religious
dogma than as a mathematical concept. 89 Moreover, the thought of
Leibniz, whose calculus represented the most genial attempt to estab-
lish a mathematics of the infinitely large and the infinitely small,
lacked resolution. Sometimes he declared against the absolute infinite,
and then again he remarked that nature, instead of abhorring the
actual infinite, everywhere made use of it to mark better the perfec-
tions of its author. 90
The symbol <*> had been used by mathematicians since the time of
Wallis to represent infinity, but no definition had been given, nor
had the Scholastic distinction been observed. The symbol was used by
Weierstrass, for example, in the sense both of a potentiality and of
an actuality: he wrote /(a) = °° to mean that -— = 0, and also
used the expression f( o° ) = b in the sense that the limit of f(x) for
x indefinitely large was b 9l To avoid this confusion, Cantor chose a
new symbol, «, to represent the actual infinite aggregate of positive
integers. It is to be remarked, furthermore, that whereas o° referred
in general to magnitude, co is to be interpreted in terms of aggregation.
Wallis, Fontenelle, and others regarded oo variously as the largest
positive integer or as the sum of all the positive integers; but the
symbol &> refers to all the positive integers only in the sense that
these form an aggregate of elements. This view of the infinite, as
concerned with groups of elements, had previously been clearly ex-
pressed by Bolzano, but he failed to recognize what Cantor called
the power of an infinite set of elements. The rational numbers can
be put into one-to-one correspondence with the positive integers, and
for this reason these two classes are regarded as having the same
power. One of the most striking results of Cantor's Mengenlehre, how-
ever, is that there are transfinite numbers higher than «• The theory
of arithmetic had shown that numbers other than the rationals were
needed for the continuum, and Cantor now showed that the desired
set of real numbers was such that these could not be put into one-to-
one correspondence with the positive integers; i. e., the set was not
89 Ibid., p. 191. . » Ibid., p. 179; see also Leibniz, Philosophised Schriften, I, 416.
91 Cajori, A History of Mathematical Notations, II, 45.
298 The Rigorous Formulation
denumerable. It therefore represented a transfinite number of a higher
power, which is often written now simply as C. Other numbers have
been found above C, but the question as to whether there is one
between a> and C is unanswered. However, the definition of the con-
tinuous variable, and hence of the concepts of the calculus, requires
only the infinite set C.
Although Cantor's work does not clear up any objection raised on
the ground of the conceptual difficulties inherent in the concept of
infinity, it definitely refutes any argument raised upon the score of
logical contradiction. Likewise, any criticism of the use of the infinite
in defining irrational number or in the limit concept is answered by
Cantor's work, which clarifies the situation, 92 As the terminus a quo
of the investigations leading to the calculus is to be found in the
Pythagorean discovery of the incommensurable and in the recognized
need for satisfactory definitions of number and the infinite, so the
terminus ad quern may be regarded as the establishment of these by
the great triumvirate: Weierstrass, Dedekind, and Cantor. The
fundamental notions of the calculus — the limit of a continuous vari-
able: the derivative and the integral — have through the work of
these men been given a logical rigor as impressive as that of Euclidean
geometry, and a formal precision of which the Greeks had never
dreamed. It has been shown that in analysis there is no need for any-
thing but whole numbers, or finite or infinite systems of these. 93
How startlingly apropos, with respect to the development of the
calculus, is the Pythagorean dictum: All is number!
82 It should, perhaps, be observed at this point that the theory of infinite aggregates
has resulted also in a number of puzzling and as yet unresolved antimonies. See Poincare,
Foundations of Science, pp. 477 ff.; Pierpont, "Mathematical Rigor," pp. 42-44. It has
been suggested, in this connection, that these paradoxes may be related to the difficulties
encountered in theoretical physics. See the review by Northrop, in Bulletin, American
Mathematical Society, XLII (1936), 791-92, of Schrodinger, Science and the Human
Temperament.
93 PoincarS, The Foundations of Science, pp. 380, 441 ff.
VIII. Conclusion
THERE is a strong temptation on the part of professional mathe-
maticians and scientists to seek always to ascribe great dis-
coveries and inventions to single individuals. Such ascription serves
a didactic end in centering attention upon certain fundamental aspects
of the subjects, much as the history of events is conveniently divided
into epochs for purposes of exposition. There is in such attributions
and divisions, however, the serious danger that too great a significance
will be attached to them. Rarely — perhaps never — is a single mathe-
matician or scientist entitled to receive the full credit for an "innova-
tion," nor does any one age deserve to be called the "renaissance" of
an aspect of culture. Back of any discovery or invention there is
invariably to be found an evolutionary development of ideas making
its geniture possible. The history of the calculus furnishes a remarkably
apt illustration of this fact. 1
The method of fluxions of Newton was no more unanticipated than
were his laws of motion and gravitation; and the differential calculus
of Leibniz had been as fully adumbrated as had his law of continuity.
These two men are to be thought of as the inventors of the calculus in
the sense that they gave to the infinitesimal procedures of their
predecessors the algorithmic unity and precision necessary for further
development. Their work differed from the corresponding methods of
their predecessors, Barrow and Fermat, more in attitude and general-
ity than in substance and detail. The procedures of Barrow and
Fermat were themselves but elaborations of the views of such men as
Torricelli, Cavalieri, and Galileo, or Kepler, Valerio, and Stevin. The
achievements of these early inventors of infinitesimal devices were in
turn the direct results of the contributions of Oresme, Calculator,
Archimedes, and Eudoxus. Finally, the work of the last-named men
was inspired by the mathematical and philosophical problems sug-
gested by Aristotle, Plato, Zeno, and Pythagoras. Without the filia-
tion of ideas which was built up by these men and many others, the
calculus of Newton and Leibniz would be unthinkable.
1 Cf. Karpinski, "Is there Progress in Mathematical Discovery?" pp. 47-48.
300 Conclusion
If, on the one hand, mathematicians have been prone to forget the
periods of suggestion and anticipation in the rise of the calculus,
historians of the subject, on the other hand, have frequently failed to
appreciate the significance of the later rigorous formulations. His-
torical accounts of the subject all too often have terminated with the
work of Newton and Leibniz, even though neither of these men was
able to furnish the precision of thought which was to follow two
centuries later. The neglect of this later period of investigation indi-
cates an inadequate attention to, or appreciation of, the fundamental
concepts of the subject as presented by Cauchy and Weierstrass.
References, which might easily have been multiplied, have frequently
been indicated above, in which the views expressed in the final elab-
oration of the calculus have most unwarrantedly been imputed to
earlier investigators in the field. Weierstrass' definition of real number
has been identified with the theory of proportion of Eudoxus, and the
Dedekind Cut with the speculations of Bryson. The continuum of
Cantor has been viewed as expressed in the speculations of William
of Occam or of Zeno. The limit concept of Weierstrass has been
interpreted as identical with the prime and ultimate ratio of Newton,
or even with the ancient Greek method of exhaustion. The derivative
and the differential of Cauchy have been described as exactly corre-
sponding to the related conceptions of Leibniz. The definite integral
of Cauchy has been ascribed in all completeness to Fermat, or even
to Cavalieri and Archimedes.
Such citations make clear how general is the tendency unguardedly
to read into the minds of earlier men one's own clear thoughts on the
subject, forgetting that these are the culmination of centuries of
speculation and investigation. The concepts of mathematics and sci-
ence are eminently cumulative in their growth — the results of a con-
tinuous effort to understand the relationships between elements and,
in terms of these, to describe the confused impressions afforded by
physical experience. The dynamics and astronomy of the sixteenth
century, for example, were not entirely new developments, but rather
grew out of medieval and ancient views on these subjects. In the same
sense the use of infinitesimal conceptions during the early modern
period did not proceed de novo, but began where the Scholastic
philosophers and Greek mathematicians had left off.
Conclusion 301
Nevertheless, the fact of such progressive achievement is not to be
interpreted as the unfolding of a well-conceived plan. Throughout the
advance of science and mathematics, elements have constantly been
discarded as well as added. For this reason no investigator is able to
foresee the direction which the elaboration of his views will take.
Only in retrospect can one trace the path along which such develop-
ment has proceeded. Although in retracing this thread of thought
one can readily recognize the notions from which the ultimate con-
cepts have sprung, the former are not in general to be identified with
the latter. Each is to be considered in the light of the mathematical
and scientific milieu of the period in which it appeared. To interpret
the geometric views of Archimedes and Barrow, for example, in terms
of the analytical symbolism of the twentieth century is tantamount to
invoking implicitly the precision and economy of thought which
modern notations afford but which these earlier investigators were far
from possessing.
A deeper and more sympathetic understanding among professional
workers in the fields of mathematics and history might easily remove
much of the misdirected thought with respect to the nature and rise
of mathematical concepts. A familiarity not only with the elements
of the calculus, but also with the history of its development, will
serve to bring out that the question is not so much who are the
founders of the subject — Weierstrass and Cauchy, or Newton and
Leibniz, or Barrow and Fermat, or Cavalieri and Kepler, or Archi-
medes and Eudoxus — but rather in what sense each of these men
may be regarded as responsible for the new analysis.
It is possible not only to trace the path of development throughout
the twenty-five-hundred-year interval during which the ideas of the
calculus were being formulated, but also to indicate certain tendencies
inimical to this growth. Perhaps the most manifest deterring force
was the rigid insistence on the exclusion from mathematics of any idea
not at the time allowing of strict logical interpretation. The very
concepts which gave birth to the calculus — those of variation and
continuity, of the infinite and the infinitesimal — were banned from
Greek mathematics for this reason, the work of Euclid being a monu-
ment to this exclusion. The work of Archimedes became most fruitful
when he abandoned the Greek logical ideal and applied such forbidden
302 Conclusion
concepts, but this represents an almost isolated example of their use.
Similarly, in the seventeenth century a number of mathematicians,
including Pascal and Barrow, avoided the use of algebra and analytic
geometry as not compatible with the demands of rigor inherited from
the ancient Greeks. Had they made full use of these elements, they
might well have been acclaimed the inventors of the calculus. Again
in the eighteenth century English mathematicians disdained, largely
because of the weak logical basis (as well as for reasons of national
jealousy), to use the differential method of the Continental "com-
putists," and consequently failed to make appreciable contributions
to the rapid growth of analysis characterizing that century.
It is clear that the indiscriminate use of methods and ideas which
are palpably without logical foundation is not to be condoned. Such
logical basis is, of course, ultimately to be sought in order to avoid
hopeless confusion (as witness the eighteenth-century use of infinite
series) ; but pending the final establishment of this, the banishment of
suggestive views is a serious mistake.
On the other hand, perhaps a more subtle, and therefore serious,
hindrance to the development of the calculus was the failure, at vari-
ous stages, to give to the concepts employed as concise and formal a
definition as was possible at the time. The paradoxes of Zeno are
excellent illustrations of the obscurity which results from a failure to
specify clearly and unambiguously the conditions of the problem and
to give formal definitions of the terms involved. Had the Greeks
demanded of Zeno a precision of statement which the mathematicians
exacted of themselves, they might not have banned the concepts lead-
ing to the calculus nor disregarded almost entirely the science of
dynamics. The nice distinctions of the Scholastic philosophers pointed
the way to the clarification of such problems, but these men were not
sufficiently familiar with the formalism of Greek geometry and Arabic
algebra to be able to carry their ideas to completion.
With the decline of Scholasticism, the tendency was away from
precision of thought and toward the free use of imagination, as found
in the literary Renaissance. The mathematical counterpart of this is
seen strongly in the works of Nicholas of Cusa and Kepler, who
employed in mathematics, without seeking adequately to define them,
the conceptions of infinity, the infinitesimal, motion, and continuity.
Conclusion 303
In a sense this was fortunate, in that it favored the development of
methods anticipating the calculus. On the other hand, however, the
lack of a sound critical attitude was to leave undefined for hundreds
of years the logical bases of the procedures thus employed, while the
resulting confusion of thought engendered half a dozen alternative
methods. Had Newton been more precise in the statement of his limit
method, and had Leibniz been more explicit in professing that he was
developing an instrument of invention and not a logical foundation,
the period of indecision might not have ensued. As it was, it required
the work of Cauchy, Weierstrass, and others to impart to the concepts
of the continuous variable, the limit, the derivative, and the integral
a precision of formulation which made them generally acceptable.
In all probability, however, the chief obstacle in the way of the
development of the concepts of the calculus was a misunderstanding
as to the nature of mathematics. Ever since the empirical mathe-
matics of the pre-Hellenic world was developed, the attitude has,
upon occasion, been maintained that mathematics is a branch either
of empirical science or of transcendental philosophy. In either case
mathematics is not free to develop as it will, but is bound by certain
restrictions: by conceptions derived either a posteriori from natural
science, or assumed to be imposed a priori by an absolutistic phi-
losophy. At the Egyptian and Babylonian level, mathematics was
largely a body of information concerning the natural world. The early
Ionians rearranged this knowledge into a deductive scheme, but the
basis was still largely empirical science. The oriental mysticism of
Pythagoras, however, reversed this state of affairs and gave to mathe-
matics a supra-sensuous reality, of which the world of appearances
was a counterpart. The premises were thus categorically established
and all the mathematician could do was to develop the logical implica-
tions of these. This view was elaborated by Plato into an idealistic
philosophy which has consistently denied the purely logical and
hypothetical nature of the propositions of mathematics. At the
Greek stage of the ideas leading to the calculus, however, the views of
Plato exerted a favorable influence, in that they counterbalanced the
Peripatetic attitude.
Aristotle considered mathematics an idealized abstraction from
natural science, and as such the premises and definitions were not
304 Conclusion
arbitrary, but were determined by our interpretation of the world of
sense perception. The concepts allowed to geometry were only such
as were consistent with this picture. The infinitesimal and the actual
infinite were excluded, not so much because of any demonstrated
logical inconsistency, but because of a supposed incompatibility with
the world of nature, from which the entities of mathematics were
regarded as derived by a disassociation of irrelevant properties. Con-
tinuity depended upon coincidence of extremities and upon oneness,
as sense perception appeared to indicate. Likewise number was
regarded, in conformity with the judgments of empiricism and com-
mon sense, as a collection of units, with the result that irrational
magnitudes were not considered as pertaining to the realm of number.
Mathematics was the logic of relations, but the nature of these was
completely determined by postulates which were in turn dictated by
the evidence of physical experience.
Of these two views, the Platonic and the Aristotelian, the former
was for a time that under which the ideas of the calculus developed.
Under this view, the conceptions of the infinite and the infinitesimal
were not excluded, inasmuch as reason was not subject to the world
of sensation. The entities of mathematics had an ontological reality,
independent of common sense, and the postulates were discovered by
reason alone. Although this made mathematics independent of natural
science, it did not give it the postulational freedom it enjoys today.
While the work of Archimedes displays elements of both the Platonic
and the Aristotelian views, it was the latter which triumphed in the
method of exhaustion and the classic geometry of Euclid. The con-
cepts of infinity and continuity were consequently, during the medieval
period, discussed from a dialectical rather than a mathematical point
of view; but when they entered into the geometry of Nicholas of
Cusa, of Kepler, of Galileo, it was largely under the Platonic view of
rational transcendentalism, rather than of naturalistic description.
Nevertheless, other mathematicians — Roberval, Torricelli, Barrow,
and Newton — were led by the natural science of the day to interpret
mathematics in terms of sense perception, as Aristotle had, and to
introduce motion to avoid the difficulties of the infinite and the con-
tinuous. The philosophers Hobbes and Berkeley felt the empirical
tendency so strongly that they denied to mathematics the idealized
Conclusion 305
concept of a point without extension, because they felt that this had
no counterpart in nature.
The attitude of most of the mathematicians of the seventeenth
century, however, was that of doubt. They employed infinitesimals
and the infinite on the assumption that they existed, and treated the
continuous as though made up of indivisibles, the results being
justified pragmatically by their consistency with Euclidean geometry.
In any case, the attitude was not that of an unprejudiced postulation
and definition, followed by logical deduction. The investigations into
the foundations of the calculus consequently took the form, during
the eighteenth century, of a search for an explanation which should
be intuitively plausible, rather than logically self -consistent. At this
time, however, there was rapidly developing a very successful alge-
braic formalism, vigorously fostered by Euler and Lagrange. This led,
in the nineteenth century, to a view of mathematics which non-
Euclidean geometry had strongly suggested — a postulational system
independent alike of the world of sense experience and of any dictates
resulting from introspection. The calculus became free to adopt its
own premises and to frame its own definitions, subject only to the
requirement of an inner consistency. The existence of a concept de-
pended only upon a freedom from contradiction in the relations into
which it entered. The bases of the calculus were then defined form-
ally in terms only of number and infinite aggregates, with no corrob-
oration through an appeal to the world of experience either possible
or necessary.
This formalizing tendency has not, however, been everywhere ac-
cepted. Even mathematicians have not always been in sympathy with
the movement. Hermite, whose favorite idea was to compare mathe-
matics with the natural sciences, was horrified by Cantor's work,
which transcended human experience. 2 Du Bois-Reymond similarly
opposed the formal definitions which were made basic in the calculus
and wished instead to define number in terms of geometric magnitude,
much as Cauchy had done implicitly, 3 thus retaining intuition as
a guide. More thoroughgoing intuitionists like Brouwer attempt
2 See Poincare, "L'Avenir des mathematiques," p. 939.
3 Pringsheim, "Nombres irrationnels et notion de limite," pp. 153 ff. See also
Jourdain, "The Development of the Theory of Transfinite Numbers," 1913-14, pp.
1, 9-10.
306 Conclusion
to visualize a fusion of the continuous and the discrete, somewhat
as Plato had. 4 The mathematician Kronecker opposed the work of
Dedekind and Cantor, not because of its formalism, but because he
thought it unnatural. These investigators had "constructed" numbers
which Kronecker felt could have no existence, and he proposed instead
to base everything on equations involving integers alone, a view which
has not been generally shared by other mathematicians. 5
A number of scientists and philosophers have naturally been even
more hesitant than these mathematicians in giving up experience and
intuition in connection with the calculus. Thoroughgoing empiricists
and idealistic philosophers in particular have sought, since the time
of Newton and Leibniz to read into the calculus a significance beyond
that of a formal postulational system. Newton had considered the
calculus as a scientific description of the generation of magnitudes,
and Leibniz had viewed it as a metaphysical explanation of such
generation. The formalism of the nineteenth century took from the
calculus any such preconceptions, leaving only the bare symbolic
relationships between abstract mathematical entities. Nevertheless,
traces of the old scientific and metaphysical tendencies remained.
Lord Kelvin, who considered mathematics the etherealization of com-
dx
mon sense, once exclaimed, when he had asked what — represented
at
v A#
and had received the answer lim — : "That's what Todhunter would
At-*o Ai
say. Does nobody know that it represents a velocity?" 6 His friend
Helmholtz showed a similar tendency. In his famous essay, Ueber die
Erhallung der Kraft, he regarded a surface as the sum of lines, 7 much
as had Cavalieri in his Exercitationes just two centuries earlier. In
another connection he asserted that incommensurable relations may
occur in real objects, but that in numbers they can never be repre-
4 See Helmholtz, Counting and Measuring, pp. xxii-xxiv; also Brouwer, "Intuitionism
and Formalism," and Simon, "Historische Bemerkungen iiber das Continuum."
5 See Couturat, De Vinfini matkematique, pp. 603 ff.; cf. also Pierpont, "Mathematical
Rigor," pp. 38-40; Pringsheim, "Nombres irrationnels et notion de limite," pp.
158-63; Jourdain, "The Development of the Theory of Transfinite Numbers," 1913-14,
pp. 2-8.
6 Hart, Makers of Science, pp. 278-79; Felix Klein, Entwicklung der Mathematik im 19.
Jahrhundert, I, 238.
7 Ueber die Erhaltung der Kraft, p. 14.
Conclusion 307
sented with exactness. 8 Mach also felt strongly the empirical origin of
mathematics and held with Aristotle that geometric concepts are the
product of idealization of physical experiences of space. 9 In conformity
with this view, he felt that some form of geometrical meaning had
necessarily to be given to the number i. 10 In this respect he is in agree-
ment with a number of present-day scientists, who feel that V — 1
simply "forms a part of various ingenious devices for handling other-
wise intractable situations." 11
The attitudes of Helmholtz and Mach are representative of the
influence in science of the positive philosophy of the nineteenth cen-
tury. Positivistic and materialistic thought were slow to accept the
changed mathematical view and insisted that the calculus be inter-
preted in terms of velocities and actual intervals, corresponding to
the data of experience and of ordinary algebra. Comte recognized that
mathematics is not "the science of magnitudes," 12 but he did not rise
to the formal view of Cauchy. In accordance with the empirical and
pragmatic attitude of Carnot, he regarded the methods of Newton,
Leibniz, and Lagrange as fundamentally identical. However, because
the differential calculus gave no clear conception of the infinitely
small and because the method of limits apparently separated the fields
of ordinary and transcendental analysis, he felt that the method of
Lagrange was to be favored. 13 More strongly expressed are the views of
Duhring, who, in 1872, in his classic Kritische Geschichte der allgemeinen
Principien der Mechanik, indulged in a polemic against Gauss, Cauchy,
and others who would deny the absolute truth of geometry, and who
would introduce into mathematics such figments of the imagination
as imaginary numbers, non-Euclidean geometry, and limits! 14 Marxian
materialists will not grant mathematics the independence of experi-
ence necessary for its proper development. 15 Such denial makes impos-
sible the concept of the derivative and the scientific description of
motion in terms thereof. The mathematical infinity is, in accord with
8 Helmholtz, Counting and Measuring, p. 26.
9 Mach, Space and Geometry, p. 94; cf. also p. 67. See also Strong, Procedures and
Metaphysics, p. 232.
xo Space and Geometry, p. 104, n. u Heyl, "The Skeptical Physicist," p. 228.
12 Comte, The Philosophy of Mathematics, p. 18. a Ibid., pp. 110-17.
14 Duhring, Kritische Geschichte der allgemeinen Principien der Mechanik, pp. 475 ff .,
529 ff.
15 Engels, Herr Eugen Diihring's Revolution in Science, p. 47.
308 Conclusion
this view, a contradiction of the "tautology" that the whole is greater
than any of its parts. 16
If a number of philosophers were led by excessive realism to reject
much of the mathematics of the nineteenth century, idealistic phil-
osophers, following Kant, were likewise unwilling to accept the bare
formalism of Cauchy and Weierstrass in the realm of the calculus.
The differential had been defined by Cauchy, not as a fixed quantity,
but as a variable, and Weierstrass had shown that the continuous
variable depends only upon the static notion of sets of elements.
Idealists attempted, nevertheless, to interpret the differential as hav-
ing an intensive quality resembling the potentiality of Aristotle, the
impetus of the Scholastics, the conatus of Hobbes, or the inertia of
modern science. They wished to view the continuum, not in terms
of the discreteness of Cantor and Dedekind, but as an unanalyzable
concept in the form of a metaphysical reality which is intuitively
perceived. The differential calculus was regarded as possessing a
"positive" meaning as the generator of the continuum, as opposed to
the "negation" of the limit concept. 17 As Hegel expressed it, the
derivative represented the "becoming" of magnitudes, 18 as opposed to
the integral, or the "has become."
Materialistic and idealistic philosophies have both failed to appre-
ciate the nature of mathematics, as accepted at the present time.
Mathematics is neither a description of nature nor an explanation of
its operation; it is not concerned with physical motion or with the
metaphysical generation of quantities. It is merely the symbolic logic
of possible relations, 19 and as such is concerned with neither approxi-
mate nor absolute truth, but only with hypothetical truth. That is,
mathematics determines what conclusions will follow logically from
given premises. The conjunction of mathematics and philosophy, or
of mathematics and science, is frequently of great service in suggesting
new problems and points of view.
u Ibid., pp. 48-49, 62; cf. also Bois, "Le Finitisme de Duhring," p. 95.
17 See Kant, Sammtliche Werke, XI (Part I), 270-71; cf. also II, 140-49 and passim.
See also Cohen, Die Princip der Infinitesimalmethode und seine Geschichte, passim; Simon,
"Zur Geschichte und Philosophic der Differentialrechnung," p. 128; Vivanti, "Note sur
l'histoire de I'infiniment petit," pp. 1 ff.; Freyer, SPudien zur Metaphysik der Differential-
rechnung, pp. 23 ff.; Lasswitz, Geschichte der Atomistik, I, 201.
M Klein, Elementary Mathematics from an Advanced Standpoint, p. 217.
18 Cohen, M. R. } Reason and Nature, pp. 171-205.
Conclusion 309
Nevertheless, in the final rigorous formulation and elaboration of
such concepts as have been introduced, mathematics must necessarily
be unprejudiced by any irrelevant elements in the experiences from
which they have arisen. 20 Any attempt to restrict the freedom of choice
of its postulates and definitions is predicated on the assumption that
a given preconceived notion of the nature of the relationships involved
is necessarily valid. The calculus is without doubt the greatest aid
we have to the discovery and appreciation of physical truth; but the
basis for this success is in all probability to be found in the fact that
the concepts involved were gradually emancipated from the qualitative
preconceptions which result from our experiences of variability and
multiplicity. Greek philosophy had attempted to separate and con-
trast the qualitative and the quantitative, but the later medieval and
early modern period associated them through geometric representation.
Even a quantitative explanation is subject to sensory notions of size,
length, duration, and so forth, so that greater independence was
achieved in the nineteenth century by basing the calculus upon
ordinal considerations only. The history of the concepts of the calculus
shows that the explanation of the qualitative is to be made through
the quantitative, and the latter is ^n turn to be explained through the
ordinal, perhaps the most fundamental notion in mathematics. As the
sensations of motion and discreteness led to the abstract notions of
the calculus, so may sensory experience continue thus to suggest prob-
lems for the mathematician, and so may he in turn be free to reduce
these to the basic formal logical/ relationships involved. Thus only may
be fully appreciated the twofold aspect of mathematics: as the lan-
guage of a descriptive interpretation of the relationships discovered
in natural phenomena, and as a syllogistic elaboration of arbitrary
premises.
20 Cf . Poincar6, Foundations of Scienci, pp. 28-29, 46, 65, 428.
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Index
Abel, N. H., 280
Academie des Sciences, 241, 242, 263
Acceleration, Greek astronomy lacked con-
cept of, 72; uniform, 82 ff., 113ff., 130,
165
Achilles, 24-25, 116, 138, 140, 295; argu-
ment in the, 24«; refuted by Aristotle, 44
Acta eruditorum, 207, 208, 214, 221, 238
A est unum calidum, 87
Agardh, C. A., Essai sur la melaphysique dn
calcul integral. 264
Albert of Saxony, 66, 68 f., 74
Alembert, Jean leRond d', 237; concept of
differentials, 246, 275; interpretation of
Newton's prime and ultimate ratio, 247;
quoted, 248; limit concept, 249, 250, 253,
257, 265, 271, 272; definition of the
tangent, 251; concept of infinitesimal,
274; concept of infinity, 275
Algebra, Arabic development, 2, 56, 60,
63 ff., 97, 120, 154; contribution of
Hindus, 63; symbols for quantities, 98;
use of, avoided in seventeenth century,
302
Algebraic formalism fostered by Euler and
Lagrange, 305
Alhazen (Ibn al-Haitham), 63
Ampere, A. M., theory of functions, 282
Anaxagoras, 40
Anaximander, 28, 40
Angle of contact, 22, 173, 174, 212
Antiphon the Sophist, 32
Apeiron, 28
Apollonius, influence in development of
analytic geometry, 187
Arabs, algebra adopted from, 2; extended
work of Archimedes, 56, 120; algebraic
development, 60, 63 ff., 97, 154
Arbogast, L. F. A., Du calcul des derivations,
263
Archimedes, 25, 29, 38, 39, 48, 49-60,
64, 65, 70, 101, 105, 112, 299,
Method, 21, 48-51, 59, 99, 125, 139,
159; axiom of, 33, 173; use of infinitely
large and infinitely small, 48 ff., 90;.
quadrature of the parabola, 49-53, 102,
124; volume of conoid, 53-55; spiral, 55-
58, 133; determination of tangents, 56 ff.;
work on centers of gravity, 60; use of
series, 76 f., 120; mensurational work, 89,
94, 107; translation and publication of
works, 94, 96, 98; modifications of
method of, Chap. IV; use of doctrine of
instantaneous velocity, 56 ff., 130; in-
equalities, 150
Area, concept of, 9, 31, 34, 58f., 62, 105,
123; application of, 18, 19, 32, 36, 96
Aristotle, 1, 20, 22, 26, 30, 46, 57, 60, 65, 66,
67, 70, 72, 79, 112, 299, 303; opposed to
Plato's concept of number, 27; On the
Pythagoreans, 37; dependence upon
sensory perception, 37 ff.; considered
mathematics an idealized abstraction
from natural science, 37-38, 303, 307;
quoted, 38, 41, 42, 43, 46; concept of the
infinite, 38ff., 69, 70, 102, 117, 151, 275,
296; concept of number, 40-41, 151, 154-
55; Physica, 41, 43, 65, 78; theory of
continuity, 41-42, 66-67, 277, 291; con-
cept of motion, 42-45, 71 ff., 176, 178,
233; denial of instantaneous velocity, 43,
73, 82; logic of, 45, 46, 47, 61; sixteenth-
century opposition to Aristotelianism, 96;
views triumphed in method of exhaustion
and geometry of Euclid, 45 ff., 304
Arithmetic triangle, see Pascal's triangle
Arrow, argument in, 24», 295; refuted by
Aristotle, 44
Assemblages, see Infinite aggregates
Athens, 26
Atomic doctrine, 21, 23, 38, 42, 66, 67, 84,
92, 180, 181
Atomism, mathematical, 22, 28, 50, 82,
115, 125, 134, 176, 188
Average velocity, 6, 8, 75, 83, 113 ff.
Babylonians, mathematical knowledge, 1,
14 ff., 62, 303; astronomical knowledge,
15
Bacon, Roger, 64; Opus majus, 66; concept
of motion, 72
Barrow, Isaac, 58, 133, 209, 299, 302, 304;
anticipated invention of the calculus, 164;
opposed arithmetization of mathematics,
175; advocated classic concept of number
338
Index
Barrow, Isaac {Continued)
and geometry, 179; distrust of algebraic
methods, 180; concepts involved in his
geometry, 181; quoted, 182; Geometrical
Lectures, 182, 185, 189, 202; method of
tangents, 182-86, 192-93, 203, 210, 214;
influence in development of the calculus,
187; concept of time, 194, 233, 295; dif-
ferential triangle, 183, 203
Bayle, Pierre, 217, 224
Beaugrand, Jean de, 120
Bede, 66
Berkeley, George, 156, 257 f.; criticism of
Newton, 224 ff.; The Analyst, 224-29,
232, 233, 235; Essay towards a New
Theory of Vision, 224; influence of
empirical tendency, 227, 304; A Defence
of Freethinking in Mathematics, 228;
Berkeley - Jurin - Robins controversy,
229 ff., 246, 263
Berlin Academy, 254
Bernoulli, James, 153, 207, 220, 238; con-
cept of infinitesimal, 239
Bernoulli, John, 207, 211, 212, 220, 238,
246; concept of infinitesimal, 239, 240;
treatment of infinite and infinitesimal,
242; concept of the integral, 278
Bhaskara, 62, 192
Binomial theorem, 119, 148, 190, 191, 194,
207, 225
Blasius of Parma (Biagio Pelicani), 84, 88,
89
Boethius, Geometry, 64
Bolzano, Bernhard, 268, 284; concept of
infinity, 116, 270, 274f., 288, 296, 297;
Rein analytischer Beweis des Lehrsatzes
. . ., 266; limit concept, 269; need for
definitions of convergence of series, 270,
280; theory of functions, 276, 282;
definition of continuity, 277, 279
Bombelli, Raphael, 97
Boyle, Robert, 2
Bradwardine, Thomas, 66fL, 97, 116; Geo-
metria speculativa, and Tractatus de con-
tinuo, 66; quoted, 67; Liber de propor-
tionibus, 69, 74
Brahmagupta, 62
Brasseur, J. B. "Exposition nouvelle des
principes de calcul differentiel," 264
Brouwer, L. E. J., 28, 67, 305; see also
Intuitionism
Bruno, Giordano, 107
Bryson of Heraclea, 32, 33, 300
Buffon, Comte de, 246, 247
Buquoy, Georg von, Eine neue Methodefiir
den Infinitesimalkalkul, 264
Buridan, Jean, 72, 113; doctrine of impetus
(or inertia), 177
Burley, Walter, 66, 74
Calculator, see Suiseth, Richard
Cambridge Platonists, 180
Cantor, Georg, 33, 271, 284; sought defini-
tion of irrational number, 288; "tiber die
Ausdehnung eines Satzes aus der Theorie
der trigonometrischen Reihen," 289;
Mengenlehre, 296, 297; concept of the
infinite, 296, 297, 298; continuum of,
296 ff., 300
Cantor-Dedekind axion, 291-92
Capella, Martianus, 66
Cardan, Jerome, 97, 118; De subtilitate, 88
Carneades, 38
Carnot, L. N. M., 226; doctrine of perfect
and imperfect equations, 158; Reflexions
sur la metaphysique du calcul infinitSsi-
mal, 257; concept of infinitesimal, 258,
283
Cauchy, A. L., Ill, 123, 268; differential
defined in terms of the derivative, 210,
275, 276; theorem of, 270, 281, 282, 289;
Cours d' 'analyse, 271; excerpt, 272, 281;
Resume des legons sur le calcul infinitesi-
mal, 271; Legons sur le calcul diffirentiel,
271; definition of limit, 272, 287; ideas
of variable, function, limit, and orders of
infinity, 272 ff., 281; concept of infinitesi-
mal, 273; definition of continuity, 277,
279
Cavalieri, Bonaventura, 28, 48, 56, 109, 151,
160, 204, 299; theorem of, 22, 118;
Geometria indivisibilibus, 111, 112, 117,
119, 121, 122, 123; forces molding
thought of, 112, 117 ff.; Exercitationes
geometricae sex, 117, 135, 306; role in
development of infinitesimal methods,
121, 123; method of indivisibles, 125, 141,
142, 150, 151, 170, 181, 195, 227
Characteristic triangle, see Differential
triangle
Circle, conception of as a limit, 32, 231,
272
Collins, John, 185
Commandino, Federigo, 99, 101, 112, 159;
Liber de centro gravitaiis solidorum, 94
Index
339
Commercium episiolicutn, 221, 222
Compensation of errors, 226, 245, 251,
257
Comte, Auguste, 307
Conatus, Hobbes's concept of, 43, 178, 179,
212, 308
Condorcet, N. C. de, 263
Continuity, concept of, 3, 4, 25, 26, 29, 35,
37, 38, 41, 44, 47, 51, 62, 63, 65-68, 116,
151, 152, 155, 179, 181, 184, 226, 229,
246, 267-70, 279, 282, 288, 291, 292, 294,
295; Aristotle's conception, 41-45, 304;
medieval conception, 65 ff., 304; law of,
93, 110, 116, 172, 178, 217-22 passim,
244, 245, 256, 258, 268, 277, 281, 283, 294,
299; Leibniz's doctrine of, 217, 268, 277,
291; definition of, 268, 277, 279, 286,
294, 298
Continuum, 3, 4, 13, 16, 17, 20, 42, 43, 45,
60, 66, 67, 115, 116, 179, 181, 217, 218,
256, 270, 291; generated by flowing of
apeiron, 28; mathematical concept of,
35, 47, 92, 227; attempted link with
indivisibles, 91; Barrow's treatment of,
180; views of Cantor imputed to earlier
investigators, 300; as an unanalyzable
concept, 308
Convergence of series, 174, 207, 253, 264,
270, 280 ff.
Cournot, A. A., Traite elementaire de la
theorie des fonctions et du calcul infinitesi-
mal, 283
Crelle's Journal, see Journal fiir die Reine
und Angewandte Malhematik
Cusa, Nicholas of, see Nicholas of Cusa
Cycloid, 151, 167
Dedekind, Richard, basis of definition of
infinite assemblages, 271; Sietigkeit und
die Irrationalzahlen, 289; theory of ir-
rational numbers, 290 ff.; concept of the
infinite, 296
Dedekind Cut, 33, 42, 292, 294, 300
Definite integral, see Integral, definite
De lineis insecabilibus, 39
Democritus, 21, 22, 26, 28, 30, 37, 39, 48,
66, 71, 91, 108; see also Atomic doctrine;
Atomism, mathematical
Derivative, concept of, 6, 13, 22, 28, 40,
43, 47, 59, 61, 63, 79, 80, 82, 85, 111, 157,
179, 184, 185, 186, 191, 196, 251, 255,
259, 262, 277, 282, 295, 298, 300, 308;
defined, 3, 4, 5, 7, 8, 11, 12, 269, 275,
293, 294; differential defined in terms of,
210, 275, 276; origin of word, 252-54;
continuous functions and, 269, 270,
279
Derived functions, Lagrange's method of,
251-55, 259-65, 267, 270, 272, 275, 278;
see also Derivative
Descartes, Rene, 2, 29, 58, 88, 133, 165,
238; Geometrie, 154, 166, 208; concept of
variability, 154, 155; method for normal
to a curve, 158; infinitesimal concept,
165, 168, 193; method of tangents, 166-
68; concept of instantaneous center of
rotation, 167; notion of instantaneous
velocity, 168, 177; Discours de la
methodc, 168; analytic geometry, 154,
165, 179, 187; number concept, 242;
method of undetermined coefficients, 166,
259
Dichotomy, 116, 295; argument in, 24w;
argument refuted by Aristotle, 44
Difference, see Differential
Differences, method of, see Differentials,
method of
Differential coefficient, 276; origin of term,
253, 265; see also Derivative
Differential quotient, 263, 275; integral the
inverse of, 283 ; see also Derivative
Differentials, 111, 123, 150, 173, 179, 187-
223, 224, 278; defined, 12, 210, 259, 275,
276, 308; method of. 59, 122, 145, 178,
185, 187-223, 224, 226, 234, 236ff., 243,
244 ff., 251, 253, 266, 267, 277, 294, 299,
300, 302; origin of word, 205-6
Differential triangle, 152, 163, 182, 203, 218,
242
Diodes, curve defined by motion, 71
Diogenes Laertius, 22
Diophantine equations, 72
Diophantus, 63; Arithmetic, 60
Distance, represented by area under
velocity-time curve, 82-84, 113-15, 126,
130-31, 134, 180
Divergent series, 207, 241, 245, 246, 265,
280
Du Bois-Reymond, Paul, 305
Diihring, Eugen; Kritische Geschichte der
allgemeinen Principien der Mechanik, 307
Duns Scotus, John, 66, 74
Dynamics, 45, 178; Galilean, 72, 82, 83,
113-15, 130, 177-78; medieval, 72 ff.,
82 ff.
340
Index
Egyptian mathematics, 1, 14ff., 21, 303
Eleatics, 23
Encyclopedia Britannica, excerpt on fluxion,
232
Epicurus, 22, 91
Eratosthenes, 48
Euclid, 1, 2, 22, 27, 30, 48, 61, 64, 90, 173;
Elements, 20, 45 ft"., 64; debt to Eudoxus,
30; quoted, 31, 33, 240; method of
exhaustion, 34, 41; logic, 46 ff.; view of
number, 47, 221; banned concepts which
gave birth to the calculus, 47-48, 301;
concept of tangent, 57
Eudoxus, 30 ff., 37, 45, 48, 299; method of
exhaustion, 30, 31, 34, 41, 46, 96; defini-
tion of proportion, 31, 47, 300
Euler, Leonhard, 167, 224, 226, 234, 250;
concept of differentials, 150, 253; func-
tion concept, 243, 272, 275; view of the
infinite, 245, 249; divergent series, 246,
280; concept of the integral, 278
Evanescent quantities, 216, 228
Exhaustion, method of, 4, 9, 26, 30, 31,
32 ff., 36, 41, 46, 48, 51 ff., 62, 95, 99, 100,
104, 106, 109, 123, 124, 128, 133, 136, 150,
171, 197, 215, 255, 258, 259, 271, 300,
304; origin of word, 34, 136, 140; concept
of infinity substituted for method of,
142
Faille, Jean-Charles della, 138
Fermat, Pierre de, 48, 56, 120, 121, 125,
127, 140, 142, 154, 164, 168, 169, 175, 187,
208, 209 f., 299, 300; method of maxima
and minima, 111, 155-59, 163, 164, 185-
86, 202; method of tangents, 157, 162 ff.,
167, 175, 183 ff., 192, 202; quadratures,
159-64, 202; use of infinite progressions,
160-61, 190; concept of definite integral,
161-62, 173; rectification of curves, 162;
differential triangle, 163, 203; Opera
varia, 164
Ferrari, Lodovico, 97
Fluent, 79, 187-223
Fluxions, 79, 173, 187-223, 224, 226, 232,
250, 260, 278; method of, 58, 59, 122, 129,
178, 187-223, 224, 225, 226, 229, 236, 237,
238, 243, 246, 247, 251, 253, 263, 265, 266,
267, 294, 299; views of Robins, Maclau-
rin, Taylor and Simpson, 232 ff.
Fonctions derivees, see Lagrange, J. L.,
method of derived functions
Fontenelle, Bernard de, 214, 241, 249;
treatment of infinite and infinitesimal,
242; EUmens de la glometrie de Vinfini,
242
Formalism, see Mathematical formalism
Forms, latitude of, 73 ff., 82, 85, 86, 87, 88,
126; graphical representation, 81-84, 111
Fourier, J. B., function concept, 276
Function concept, 55, 56, 58, 60, 94, 98,
156, 184, 186, 196, 220, 221, 236, 237,
240, 243, 259, 272, 273, 276
Fundamental theorem of the calculus, 10,
11, 164, 181, 184, 187, 191, 194, 196, 203,
204 ff., 279
Galileo, 2, 29, 70, 121, 177, 180, 299; con-
cept of the infinite, 70, 115 ff., 193, 270;
dynamics, 72, 82, 83, 89, 113 ff., 133;
recognition of law of uniform accelera-
tion, 82, 85, 113ff., 130; concept of area,
84, 134; Two New Sciences, 85, 112, 113,
115; forces molding thought of, 112 ff.;
idea of impetus, 113, 130, 177; concept of
the infinitely small, 115 ff., 195; doctrine
of continuity, 116, 291
Gassendi, Pierre, 224
Gauss, K. F., 98, 268, 280
Geometry, Greek, see Greeks
Gerdil, Hyacinth Sigismund, 250
Girard, Albert, 104
Goethals, Henry, 66, 74
Grandi, Guido, 218; interest in dif-
ferential calculus, 238; concept of in-
finitesimal, 241
Graphical representation of variables, 80 ff.,
Ill, 113ff., 190, 288, 309
Gravity, centers of, 60, 93, 94, 99 ff., 104,
139, 145, 158, 163, 166
Greeks, search for universals, 8, 14; devel-
opment of mathematics and science,
16 ff.; geometry, 17, 26, 46, 65; concept of
number, 18, 29, 41, 43; position of Zeno's
paradoxes in thought of, 25; concept of
length, area, and volume, 31 ff.; concept
of proportion, 31 ff.; method of exhaus-
tion, 33S., 35, 100, 136, 197, 271, 300;
concept of motion, 42 ff., 71, 132;
skeptics, 46, 214; abandonment of at-
tempt to associate numbers with all
geometric magnitudes, 20, 62; mathe-
matics in Middle Ages, 64; concept of
instantaneous velocity, 43, 57, 82
Index
341
Gregory, James, 192, 280; Vera circuit et
hyperbolae quadratura, 174; arithmetical
and analytical work, 175; quadratures,
175; use of Fermat's tangent method,
175, 183
Gregory of Rimini, 66, 69
Gregory of St. Vincent, 162, 202; Opus
geometricum, 135, 160, 169; "ductus
plani in planum," 135, 145; view of
nature of infinitesimals, 135 ff.; use of
word "exhaust," 136; limit concept, 137,
152, 169, 282; geometric progressions,
137-38, 174, 190; influence, disciples, 138;
method of indivisibles, 141
Grosseteste, Robert, 66
Guldin, Paul, 121, 138
Guldin's Rule, see Pappus, theorem
Hadamard, J., 276
Halley, Edmund, 222, 224
Hankel, Hermann, 271
Harmonic triangle, 204
Harriot, Thomas, 169
Hegel, G. W. F., 308
Heine, Eduard, 290; limit concept, 287;
"Die Elemente der Funktionenlehre,"
288
Helmholtz, Hermann von, 294, 307; Die
Erhaltung der Kraft, 306
Hentisbery, William of, 84, 88, 96, 113,
177
Heraclitus, 25, 71
Hermite, Charles, 305
Heron of Alexandria, 59, 115
Heterogenea, 140, 152
Heuraet, Heinrich van, rectification of
curves, 162
Hindus, Pythagoras' debt to, 19; mathe-
matics, 60, 61 ff., 72, 97
Hipparchus, 59, 63
Hippias of Elis, 56, 71
Hippocrates of Chios, 33
Hobbes, Thomas, 212, 227; concept of the
conatus, 43, 178, 179, 195, 308; opposed
arithmetization of mathematics, 175;
concept of number and of the geometrical
elements, 176; idea of motion, 178
Homogenea, 140
Horn angle, see Angle of contact
Hudde, Johann, rules for tangents and for
maxima and minima, 185 f.
Humanism, 89
Huygens, Christiaan, 153, 208, 213; use of
Fermat's tangent method, 183, 186; rules
for tangents and for maxima and minima,
185 f.
Ibn al-Haitham, see Alhazen
Impetus (inertia), concept of, 43, 72, 130,
177
Incommensurable, see Irrational
Indefinite integral, see Integral, indefinite
Indivisibles, 11, 13, 22, 27, 38, 63, 66 ff.,
84,91, 109, 116, 118ff., 123, 124, 151, 173,
197, 227; method of, 50, 56, 92, 117-26,
133, 134, 139, 140, 141, 142, 143 ff., 149 ff.,
181
Inertia, see Impetus
Infinite aggregates, 13, 25, 68, 70, 115, 249,
257, 268, 270, 275, 277, 284, 286, 290, 291,
293-98; definition, 296
Infinitely small, see Infinitesimal
Infinite sequence, 4, 7, 8, 9, 11, 24, 36, 37,
50, 53, 102; see also Infinite series
Infinite series, 24, 44, 52, 55, 56, 76, 86, 116,
126, 137, 140, 142, 152, 154, 173, 190,
192, 199, 216, 221, 226, 231, 245, 254,
256, 264, 265, 267, 268, 270, 276, 280,
281, 282, 284, 286, 289, 290, 302; con-
vergence of, 174, 207, 253, 264, 270,
280 ff.
Infinitesimal, 7, 11, 12, 22, 29, 50, 51, 55,
59, 62, 63, 65, 68, 83, 84, 90, 92, 96, 108,
109, 114, 115, 117, 135, 142, 145, 152,
162, 163, 166, 169, 176, 180, 181, 183 ff.,
187, 191, 193, 195, 196, 197, 199, 200,
201, 202ff., 209ff., 212, 215, 218, 220,
224, 225, 229 ff., 258, 260, 263, 265, 286,
299, 301; concept of, 4, 21, 28, 39, 47,
154 ff., 213 ff., 239 ff., 262, 300 {see also
under Cavalieri; Descartes; Pascal); not
found before Greek period, 15; Zeno's
dictum against, 23; Democritean view,
30; Aristotelian view, 38 ff.; as an in-
tensive quantity, 43, l78f., 262; Archi-
medes' view, 49 ff.; medieval views, 66 ff.;
potential existence, 67; view of Nicholas
of Cusa, 90 ff.; Kepler's view, 108 ff.;
Galileo's view, 112 ff.; view of Gregory of
St. Vincent, 135 ff.; Tacquet's view,
139 ff.; Roberval's view, 141 ff.; Pascal's
view, 149 ff.; Fermat's view, 154 ff.;
Wallis' view of, 170ff.; philosophers
reluctant to abandon, 179; Barrow's
342
Index
Infinitesimal (Continued)
view, 180 ff.; Newton's view, 193 ff.;
Leibniz's view, 209 ff.; confused with
fluxions, 223; Jurin's view, 229; Robins'
disavowal of, 230; views of John and
James Bernoulli, 238 ff.; views of Wolff
and Grandi, 240-41; view of Fontenelle,
241-42; Euler's view, 244-46; definition,
248, 273; D'Alembert's view, 248f.; views
of Kastner and Lagrange, 250 ff.; Car-
not's view, 258 f.; views of Bolzano and
Cauchy, 270 ff.
Infinitesimals, fixed, see Indivisibles
Infinity, 2, 6, 22, 38, 41, 44, 51, 52, 56, 63,
65, 66, 70, 76, 77, 90, 91, 92, 108, 121,
143, 151, 152, 172, 213, 218, 239ff., 267;
concept of, 25, 39, 47, 53, 115ff., 226,
268, 296, 302, 304; Aristotle's view, 38-
41, 68; actual and potential, 40, 41,' 68,
69, 77, 102, 117, 151; banned from Greek
mathematics, 46, 47, 96, 142, 301;
Scholastic view, 68 ff.; Galileo's view, 70,
115, 193, 270; mathematical concept, 116,
170-71, 227, 270-71, 274-75, 284, 296ff.;
Bolzano's view, 116, 270-71, 296; sym-
bols used to represent, 170, 297; orders of,
245, 274; Cauchy's view, 274-75, 284,
296; view of Dedekind and Cantor, 296 ff.
Inscribed and circumscribed figures, propo-
sition of Luca Valerio, 124, 128, 137
Instantaneous center of rotation, 167
Instantaneous direction, 57f., 132, 134, 146,
189, 212
Instantaneous rate of change, 78, 81, 82, 83
Instantaneous velocity, 6, 7, 8, 43, 44, 47,
59, 73, 82, 84, 115, 116, 130 ff., 168, 178,
179, 180, 191, 193 ff., 200, 221, 227, 232,
233, 295; quantitative treatment, 73 ff.,
82 ff., 113 ff., 130 ff., 177; medieval view,
73 ff.; Barrow's treatment, 180; Newton's
view, 194 ff.; views of Robins, Mac-
laurin, Taylor, and Simpson, 234
Instants, 92
Integral, 22, 40, 80, 109, 123, 161, 187-223
293, 294, 298; defined, 3, 5, 11, 12, 206,
239, 278, 279, 294; concepts of, 6, 13, 47,
56, 61, 63, 184, 185, 295, 308; origin of
word, 67, 205, 206
Integral, definite, 8, 50, 56, 120, 121, 123,
127, 128, 143, 144, 146, 149, 152, 159,
161, 191, 206, 278; defined, 9, 10, 55, 279;
concept of, 142, 173
Integral, indefinite, 191, 206, 278; defined, 1 1
Integral of Lebesque, 280
Integration, by parts, 152, 163
Intensity, average, 74-78
Intensive magnitude, 28, 43, 177, 178, 179,
262, 308
Intuition, 5; Euclid's Elements based on, 47;
of uniform rate of change, 78
Intuitionism, mathematical, 3, 28, 67, 209,
305
Ionians, 18, 21; empirical science the basis
of mathematics of, 303
Irrational and incommensurable, 18 ff., 20,
31 ff., 35, 37, 61 ff., 66, 96, 97, 107, 121,
173, 174, 176, 179, 190, 256, 281, 282,
284, 285, 288-94, 296, 300, 306
Isidore of Seville, 66
James of Forli, 74
John XXI, pope (Petrus Hispanus), Sutn-
mulae logicales, 68
Jordanus Nemorarius, 64, 96, 98, 173
Journal des savants, 238
Journal fiir die Reine und Angewandte
Mathematik, 288
Jurin, James, Geometry No Friend to In-
fidelity, 228; The Minute Mathematician,
228; attitude toward infinitesimal, 229;
controversy with Berkeley and Robins,
229 ff., 246, 263
Kant, Immanuel, 2, 3, 261, 264, 308
Kastner, A. G., 254; Anfangsgriinde der
Analysis des Unendlichen, 250
Kelvin, William Thomson, baron, quoted,
306
Kepler, Johann, 2, 48, 91, 94, 156, 299;
influenced by Nicholas of Cusa, 93;
speculative tendency, 106; Mysterium
cosmographicum, 107; work on curvilinear
mensuration, 108; Nova stereometria
doliorum, 108, 110, 111, 119; Astronomia
nova, 109; theory of maxima and mini-
ma, 110; static approach to calculus, 111;
forces molding thought of, 112; view of
infinity, 117
Kronecker, Leopold, opposed work of
Dedekind and Cantor, 306
La Chapelle, de, Institutions de gtomitrit,
247
Lacroix, S. F., 253; Traite du calcul differen-
tiel et du calcul integral, 264; Traite" 6l6-
Index
343
tnentaire, 265; limit concept, 271, 272;
concept of the integral, 278; employment
of divergent series, 280
Lagrange, J. L., 224, 226, 251; quoted, 165,
252; method of derived functions, 251-55,
259-65, 267, 270, 272, 275, 278; concept
of convergent series, 253, 270, 280;
Thiorie des fonctions analytiques, 260-61,
263
Landen, John, The Residual Analysis, 236
Laplace, Pierre Simon, Marquis de, 224
Latitude of forms, see Forms, latitude of
Law of continuity, see Continuity
Law of uniformly difform variation, see
Uniform acceleration
Lebesque, integral of, 280
Leibniz, Gottfried Wilhelm von, 4, 8, 10,
13, 29, 47, 48, 59, 67, 82, 145, 153, 162,
187-223, 230, 237, 287, 299, 300; infini-
tesimal method, 28, 158, 210, 213, 219,
243, 265, 273, 274; reference to Calcula-
tor, 88; sought basis of calculus in genera-
tion of magnitudes, 94; differential
method, 122, 178, 219, 239, 244, 247, 275,
294; views influenced by Pascal, 150;
concept of integral as totality, 173; dif-
ferential triangle, 182, 203; development
of algorithmic procedure, 185, 188, 202;
definition of the integral, 206, 278; "A
New Method for Maxima and Minima,"
207; definition of first-order differentials,
210; Historia et origo calculi differ entialis,
215; quoted, 217, 218; Theodicie, 218;
view of the infinite, 219, 297; priority
claims of Newton and, 188, 221, 222, 246;
law of continuity (see under Continuity)
Leibniz's Rule, 252
Length, concept of, 19, 31, 58 f.
Leonardo da Vinci, 2, 88; influenced by
Scholastic thought, 92; center of gravity
of tetrahedron, 93; use of doctrine of
instantaneous velocity, 130; idea of
impetus familiar to, 177
Leonardo of Pisa, 65, 71, 97; Liber abaci, 64
Leucippus, 66
Lever, law of the, 49
L'Hospital, Marquis de, 226; Analyse des
infinimenls petits, 238, 241
L'Huilier, Simon, Exposition ilimenlaire des
principes des calctds superieurs, 255;
definition of differential quotient, 255;
limit concept, 257, 265, 271, 272; view of
differentials, 275; view of the integral, 278
Limit, 15, 25, 34, 37, 50, 51, 52, 53, 62, 109,
121, 126, 158, 174, 180, 197, 198, 200, 201,
211, 213, 214, 218, 220, 226, 255, 260;
definition, 7, 8, 36, 272, 287; derivative
based on the idea of, 7, 8, 279, 293, 294;
concept of, 24, 27, 30, 32, 43, 53 ff., 58,
73, 77, 79, 85, 87, 91, 98, 102 ff., 106, 114,
117, 123, 124, 132, 134, 137, 142, 153, 156,
167, 169, 173, 176, 179, 180, 181, 195, 198,
216, 218, 222, 228, 233, 235, 244ff., 262,
266, 271 ff., 281, 282, 284, 286 ff., 295, 298;
method of, lOOff., 117, 146, 150, 157, 229,
240, 263, 264, 267, 284; doctrine of, 140;
controversy between Robins and Jurin,
230; calculus interpreted in terms of,
247 ff., 254 ff., 267 ff.; involved definition
of number, 290, 296
Lines, incommensurability of, 19, 20; as
velocitiesor moments, 81 ff., 114, 115, 117
Lucretius, 68
Mach, Ernst, view of continuity, 291, 294;
geometric concepts, 307
Maclaurin, Colin, 82, 234-35; concepts of
time and instantaneous velocity, 233;
Treatise of Fluxions, 233, 260
Major, John, 85
Marsilius of Inghen, 84
Mathematical atomism, see Atomism,
mathematical
Mathematical formalism, 3, 67, 209, 243,
246, 253, 272, 283, 284, 286, 305
Mathematical Gazette, 276
Mathematische Annalen, 289
Maurolycus, Franciscus, 159
Maxima and minima, 16, 85, 110, 155-59,
163, 164, 185, 192, 208
Mengenlehre, 296, 297; see also Infinite
aggregates
Meray, Charles, Nouveau precis d'analyse
infinitesimale, 288, 289; "Remarques sur
la nature des quantitfe d6finies," 289
Mersenne, Marin, 165, 166
Middle Ages, 2, 60, 61-95; speculations on
infinite, infinitesimal, and continuity, 94;
reaction against work of the, 96
Miscellanea Taurinensia, 254
Moments, 150, 178, 191, 193, 194, 195, 198,
212, 221, 236; lines as, 114, 115, 117;
Newton's use of, 122, 229, 235; confusion
between fluxions and, 224, 232
Monad, 23, 28
344
Index
Montucla, Etienne, 151
Motion, nature of, 6-7, 42, 294-95; in-
stantaneous direction of, 57, 130 ff., 146-
47, 166-67; uniformly accelerated, 82 ff.,
113 ff., 130, 165; graphical representation
of, 80 ff.
Napier, John, 122
Neil, William, rectification of curves, 162
Neoplatonism, 45, 177
Newton, Sir Isaac, 4, 10, 13, 28, 29, 47, 48,
51, 59, 82, 116, 150, 162, 174, 180, 187-
223, 236, 268, 277, 278, 299, 300; method
of fluxions, 58, 79, 82, 122, 129, 133, 178,
190, 213, 220, 221, 225, 229, 278, 294;
view of moments, 122, 235; concept of
limit, 145, 271; invention of algorithmic
procedures, 185, 188; concept of number,
190, 242; De analysi, 190, 194, 200, 202,
228, 235; use of infinite series, 190; use of
infinitely small, and binomial theorem,
191, 194; Methodus fluxioniim, 193, 194,
195, 196, 200, 206, 246; definition of
fluent and fluxion, 194, 206; De quadra-
ture, 195, 196, 197, 200, 201, 202, 205,
206, 221, 222, 225, 228, 247; Principia,
197, 198, 199, 200, 201, 205, 220, 222, 225,
228, 229, 271; concept of infinitesimal,
226, 274; renunciation of infinitesimals,
213, 222;"priority claims of Leibniz and,
221, 222, 246; prime and ultimate ratios,
222, 225, 229 ff., 254, 259, 287, 294, 300;
Berkeley's criticism of, 224 ff.; need for
clarification of terms used by, 224, 228,
230; view of continuity of time, 194, 295
Nicholas of Cusa, 2, 88, 89 ff., 93, 107, 108,
135; quadrature of the circle, 91; views
on infinite and infinitesimal, 92, 93; idea
of continuity, 110; influence, 113; idea of
impetus familiar to, 177; influenced by
Platonic view, 88-90, 107, 304
Nicomachus, 143-44
Nicomedes, 71
Nieuwentijdt, Bernard, 213, 224, 248
Nondifferentiable continuous function, 269,
270, 276, 282, 285, 288
Notations of the calculus, 191, 193-95, 198,
205, 220, 223, 252, 265, 268, 275
Number, concept of, 15, 29, 30, 31, 36, 41,
42, 46, 51, 52, 60, 61, 62, 63, 71, 97, 154,
174, 176, 178, 179, 180, 190, 216, 221,
226, 231, 242, 256, 270, 273, 281, 282, 284,
285, 286, 289-94, 296, 298, 304, 305, 306;
Pythagorean view, 18 ff.; Greek view, 20,
40-45; Plato's view, 27
Numerals, Hindu-Arabic, 63, 71
Oldenburg, Henry, 196
Oresme, Nicole, 65, 79 ff., 84, 96, 177, 299;
Tractatus de latitudinibus formarum, 80,
85, 88; Tractatus de figuratione potentia-
rum et mensurarum, 81 ; idea of instan-
taneous velocity, 82 ff.; consideration of
infinite series, 86; study of maxima and
minima, 85-86, 111, 156; concept of
uniform acceleration, 82-83, 113, 130;
use of geometric representations, 80 ff.,
125, 154; influence in development of
analytic geometry, 81 ff., 154, 165, 187;
infinitesimal concept, 84 ff., 114 ff., 193
Oughtred, William, 170
Oxford school, 74, 80, 84, 87
Pacioli, Luca, 88; Sumtna de arithmetica, 64,
97
Pappus of Alexandria, 155; theorem, 60, 63,
108, 139
Parabola, quadrature of the, 49-53, 159 ff.
Paracelsus, 177
Parallelogram of velocities, 57, 129-34, 146
Paris School, 80, 84, 87
Parmenides, 23
Pascal, Blaise, 56, 140, 142, 150, 160, 162,
168, 209; concept of infinitesimal, 121,
147, 148, 150, 151, 239; Potestatum
numericarum sumtna, 148; TraitS des
sinus du quart de cercle, 151, 153, 203;
avoided the use of algebra and analytic
geometry, 152-53, 302
Pascal, Etienne, 147
Pascal's triangle, 148, 149, 204
Pericles, 26
Petrus Hispanus, see John XXI, pope
Physis, 21
Plato, 1, 21, 25, 26, 29, 30, 37, 39, 45, 46, 48,
89, 90, 91, 93, 291, 299, 303; concept of
mathematics, 1, 26 ff., 60, 303, 304;
criterion of reality, 28; influence, 89-90,
93, 107
Plutarch, 22; quoted, 23
Poincare, Henri, 13
Poisson, S. D., Traiti de mecanique, 283
Positivism, 5, 307
Index
345
Prime and ultimate ratios, 116, 195 ff., 213,
216ff., 221, 250, 256, 262, 267; probable
origin of term "ultimate ratio," 197;
Newton's method, 222, 225, 229 ff., 254,
259, 287, 294, 300; see also Number,
concept of
Problemum austriacum, see Gregory of St.
Vincent, Opus geometricum
Proclus, 20, 46; quoted, 17
Proportion, concept of, 30 ff., 47
Ptolemy, 59
Pythagoras, 17, 18, 89, 299, 303; debt to
Hindus, 19
Pythagoreans, 1, 18 ff., 26, 39; mathemati-
cal concepts, 17 ff., 18, 40, 176; applica-
tion of areas, 18 ff., 32, 62, 96; problem of
the incommensurable, 18 ff ., 61 ; influence,
89, 93, 107, 115, 143
Quadratures, 49-53, 91, 104f., 108, 119ff.,
124 ff., 144 ff., 149f., 159 ff., 170 ff., 191 ff.,
Quartic, solved, 97
Ramus, Petrus, 96
Raphson, Joseph, The History of Fluxions,
222
Ratios, see Number concept; Prime and
ultimate ratios
Rectification of curves, 133, 162, 163
Regiomontanus, 92
Riemann, G. F. B., 123
Roberval, Giles Persone de, 56, 58, 125, 133,
140ff., 151, 152, 160, 162, 168, 177, 209,
304; concept of infinitesimal, 121, 141 ff.;
Traits des indivisibles, 141, 147, 152;
method of indivisibles, 141 ff., 150;
association of numbers and geometrical
magnitudes, 142; obscured limit idea
through notion of indivisibles, 144;
arithmetizing tendency in quadratures,
142 ff., 168; method of tangents, 146
Robervallian lines, 145, 175
Robins, Benjamin, controversy with Berke-
ley and Jurin, 229ff., 246, 263; A Dis-
course Concerning . . . Newton's Methods,
229; limit concept; 230 ff., 271; dis-
avowal of infinitesimals, 230; criticism of
Euler, 246; interpretation of Newton's
prime and ultimate ratio, 247
Rolle, Michel, 241
Royal Society, 221
Russell, Bertrand, 3n, 67; definition of real
number, 293
Scholastics, 2, 4, 26, 40, 60, 61-95; view of
impetus, 43, 72, 177; discussions of the
infinitely large and small, 66 ff., 77, 90;
categorematic and syncategorematic in-
finities, 68ff., 115, 170, 296-97; interest
in dynamics, 72 f., 82 ff., 133; concept of
instantaneous velocity, 73, 82; considera-
tion of infinite series, 76 ff., 86 ff., 190;
geometrical representation of variables,
80 ff., Ill; knowledge of uniformly
accelerated motion, 82 ff . ; reaction against
methodology of, 96; influence upon
Galileo, 112
Sensory perception, limitations, 6, 37, 38,
39, 43, 295
Servois, F. J., Essai sur un nouveau mode
^exposition des principes du calcul dif-
ferentiel, 263
Simplicius, 22
Simpson, Thomas, view on instantaneous
velocities, 234
Skeptics, Greek, 46, 214
Sluse, Ren6 Frangois de, 162, 185
Snell, Willebrord, 109; Tiphys Batavus,
152
Sniadecki, J. B., 261
Soto, Dominic, 85
Spiral, of Archimedes, 55 ff., 133
Stade, 295; argument in the, 24n; argument
refuted by Aristotle, 44
Stevin, Simon, 91, 104, 108, 123, 130, 145,
152, 299; propositions on centers of
gravity, 99 ff.; limit idea, 102, 137, 169;
influence, 141
Stifel, Michael, 93, 97, 108
Strato of Lampsacus, 39
Suiseth, Richard (Calculator), 83, 88, 96,
113, 115, 130, 177, 299; Liber calcula-
tionum, 69, 74, 87, 88; concept of varia-
bility, 74 ff., consideration of infinite
series, 76 ff., 86; use of words fluxion and
fluent, 79, 194
Symbols, for quantities in algebra, 98; used
to represent infinity, 170, 297; used in
calculus, 194, 205, 253
Tacquet, Andreas, 138; Cylindricorum et
annularium, 139, 151; Arithmeticae theo-
ria et praxis, 140; method of indivisibles,
140, 151, 181; limit concept, 140, 152,
169, 230; propositions on geometric
progressions, 140, 174, 190
346
Index
Tangent, Greek concepts, 56 ff.; Archi-
medes' method, 57 f.; Torricelli's method,
128-33, 146, 157; definition, 129, 132,
166, 174, 249; Roberval's method, 146-
47; Fermat's method, 157-58, 162-64,
167, 175, 183, 192-93, 202, 203; method
of Descartes, 166; Barrow's method, 182-
86, 192-93
Tartaglia, Nicolo, 94, 97
Taylor, Brook, 235, 244-45; Methodus
incrementorum, 234
Taylor's series, 261, 267
Thales, 1, 16, 17, 28
Theophrastus, 39
Thomas, Alvarus, 85 ; Liber de triplici motu,
87
Thomson, William, see Kelvin, William
Thomson, baron
Time, concept of, 180, 194, 233, 295
Todhunter, Isaac, 306
Torricelli, Evangelista, 56, 58, 59, 122,
123 ff., 134, 145, 160, 162, 168, 177, 299,
304; view of infinitesimals, 121; De
dimensione parabolae, 124; use of method
of indivisibles, 125, 133, 134; De infinitis
hyperbolis, 127, 152; method of tangents,
128 ff., 146, 157; method of exhaustion,
128, 133; concepts of motion and time,
130 ff., 168, 177, 180; differential triangle,
203
Transfinite number, 298
Triangle, arithmetic, see Pascal's Triangle
Triangle, differential, 152, 163, 182, 203,
218, 242; harmonic, 204
Ultimate ratio, see Prime and ultimate ratio
Uniform acceleration, 82 ff., 113ff., 130, 165
Uniform intensity, concept of, 83
Uniformly difform variation, law of, see
Uniform acceleration
122, 154ff., 167, 176 ff., 180, 193 ff., 273,
286, 288, 294, 301; see also Graphical
representation of variables; and Forms,
latitude of
Variation, law for uniformly difform, see
Uniform acceleration
Varignon, Pierre, 241, 280
Velocities, parallelogram of, 57, 129-34,
146; see also Instantaneous velocity
Venatorius, 94
Viete, Francois, 93, 98, 108; influence in
development of analytic geometry, 154,
187
Volume, concept of, 31, 34, 58, 123
Wallis, John, 56, 59; treatment of infinite
and infinitesimal, 121, 170, 239, 242;
Arithmetica infinitorum, 140, 170, 171,
176, 189; applied analytic geometry to
quadratures, 168f.; limit concept, 169;
De sectionibus conicis tractatus, 170, 171;
concept of definite integral, 173; interest
in angle of contact, 173; arithmetization
criticized, 179; Algebra, 201
Weierstrass, Karl, 31; continuous nondif-
ferentiable function, 270, 282, 288;
trend toward formalism, 283; limit con-
cept, 284, 286-88, 300; concept of
irrational number, 285-300 passim; con-
cept of the infinite, 288, 296
Whewell, William, 256
William of Hentisbery, see Hentisbery,
William
William of Occam, 66, 67, 69, 300
Wolff, Christian, 238; concept of infinitesi-
mal, 240; number concept, 243
Wren, Sir Christopher, 162, 169
Wronski, Hoen6, differential method, 261;
Refutation de la Morie des fonctions
analytiques, 261n, 264; limit concept, 262
Valerio, Luca, 104 ff., 108, 112, 123, 299;
De centro gravitatis solidorum, 104;
proposition on inscribed and circum-
scribed figures, 105, 124, 128, 137; limit
concept, 106, 230
Variability, concept of, 4, 24, 25, 42 ff., 47,
51, 55, 56, 60, 71 ff., 78 ff., 92, 94, 96, 98,
Xenocrates, 39
Xenophanes of Colophon, 23
Zeno, 43, 267, 272, 299, 300; paradoxes, 4,
8, 23 ff., 37, 44, 53, 62, 77, 116, 138, 140,
179, 231, 281, 287, 288, 295, 302
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21851-1 Paperbound $2.75
The I Ching (The Book of Changes), translated by James Legge. Complete
translated text plus appendices by Confucius, of perhaps the most penetrating divina-
tion book ever compiled. Indispensable to all study of early Oriental civilizations.
3 plates, xxiii + 448pp. 21062-6 Paperbound $3.00
The Upanishads, translated by Max Miiller. Twelve classical upanishads: Chan-
dogya, Kena, Aitareya, Kaushitaki, Isa, Katha, Mundaka, Taittiriyaka, Brhadaran-
yaka, Svetasvatara, Prasna, Maitriyana. 160-page introduction, analysis by Prof.
Miiller. Total of 826pp. 20398-0, 20399-9 Two volumes, Paperbound $5.00
CATALOGUE OF DOVER BOOKS
Algebras and Their Arithmetics, Leonard E. Dickson. Complete background
for advanced study of abstract algebra. Clear rigorous exposition of the structures
of many special algebras, from an elementary introduction to linear transformations,
matrices and complex numbers to a generalization of the classic theory of integral
algebraic numbers. Each definition and theorem illustrated by a simple example.
xii + 241pp. 60616-3 Paperbound $1.50
Astronomy and Cosmogony, Sir James Jeans. Modern classic of exposition,
Jean's latest work. Descriptive astronomy, atrophysics, stellar dynamics, cosmology,
presented on intermediate level. 16 illustrations. Preface by Lloyd Motz. xv +
428pp. 60923-5 Paperbound $3.50
Experimental Spectroscopy, Ralph A. Sawyer. Discussion of techniques and
principles of prism and grating spectrographs used in research. Full treatment of
apparatus, construction, mounting, photographic process, spectrochemical analysis,
theory. Mathematics kept to a minimum. Revised (1961) edition. 110 illustra-
tions, x -f- 358pp. 61045-4 Paperbound $3.00
Theory of Light, Richard von Mises. Introduction to fluid dynamics, explaining
fully the physical phenomena and mathematical concepts of aeronautical engineer-
ing, general theory of stability, dynamics of incompressible fluids and wing theory.
Still widely recommended for clarity, though limited to situations in which air
compressibility effects are unimportant. New introduction by K. H. Hohenemser.
408 figures, xvi + 629pp. 60541-8 Paperbound $3.75
Airplane Structural Analysis and Design, Ernest E. Sechler and Louis G.
Dunn. Valuable source work to the aircraft and missile designer: applied and
design loads, stress-strain, frame analysis, plates under normal pressure, engine
mounts, landing gears, etc. 47 problems. 256 figures, xi -\- 420pp.
61043-8 Paperbound $2.50
Photoelasticity: Principles and Methods, H. T. Jessop and F. C. Harris.
An introduction to general and modern developments in 2- and 3 -dimensional stress
analysis techniques. More advanced mathematical treatment given in appendices.
164 figures, viii + 184pp. 6y 8 x 9^4- (USO) 60720-8 Paperbound $2.00
The Measurement of Power Spectra From the Point of View of Com-
munications Engineering, Ralph B. Blackman and John W. Tukey. Techniques
for measuring the power spectrum using elementary transmission theory and theory
of statistical estimation. Methods of acquiring sound data, procedures for reducing
data to meaningful estimates, ways of interpreting estimates. 36 figures and tables.
Index, x + 190pp. 60507-8 Paperbound $2.50
Gaseous Conductors: Theory and Engineering Applications, James D.
Cobine. An indispensable reference for radio engineers, physicists and lighting
engineers. Physical backgrounds, theory of space charges, applications in circuit
interrupters, rectifiers, oscillographs, etc. 83 problems. Over 600 figures, xx +
606pp. 60442-X Paperbound $3.75
CATALOGUE OF DOVER BOOKS
Adventures of an African Slaver, Theodore Canot. Edited by Brantz Mayer.
A detailed portrayal of slavery and the slave trade, 1820-1840. Canot, an established
trader along the African coast, describes the slave economy of the African kingdoms,
the treatment of captured negroes, the extensive journeys in the interior to gather
slaves, slave revolts and their suppression, harems, bribes, and much more. Full and
unabridged republication of 1854 edition. Introduction by Malcom Cowley. 16
illustrations, xvii + 448pp. 22456-2 Paperbound $3.50
My Bondage and My Freedom, Frederick Douglass. Born and brought up in
slavery, Douglass witnessed its horrors and experienced its cruelties, but went on
to become one of the most outspoken forces in the American anti-slavery movement.
Considered the best of his autobiographies, this book graphically describes the in-
human treatment of slaves, its effects on slave owners and slave families, and how
Douglass's determination led him to a new life. Unaltered reprint of 1st (1855)
edition, xxxii + 464pp. 22457-0 Paperbound $2.50
The Indians' Book, recorded and edited by Natalie Curtis. Lore, music, narratives,
dozens of drawings by Indians themselves from an authoritative and important
survey of native culture among Plains, Southwestern, Lake and Pueblo Indians.
Standard work in popular ethnomusicology. 149 songs in full notation. 23 draw-
ings, 23 photos, xxxi + 584pp. 6% x 9%. 21939-9 Paperbound $4.50
Dictionary of American Portraits, edited by Hayward and Blanche Cirker.
4024 portraits of 4000 most important Americans, colonial days to 1905 (with a
few important categories, like Presidents, to present) . Pioneers, explorers, colonial
figures, U. S. officials, politicians, writers, military and naval men, scientists, inven-
tors, manufacturers, jurists, actors, historians, educators, notorious figures, Indian
chiefs, etc. All authentic contemporary likenesses. The only work of its kind in
existence; supplements all biographical sources for libraries. Indispensable to any-
one working with American history. 8,000-item classified index, finding lists, other
aids, xiv + 756pp. 9\/ 4 x 12%. 21823-6 Clothbound $30.00
Tritton's Guide to Better Wine and Beer Making for Beginners, S. M.
Tritton. All you need to know to make family-sized quantities of over 100 types
of grape, fruit, herb and vegetable wines ; as well as beers, mead, cider, etc. Com-
plete recipes, advice as to equipment, procedures such as fermenting, bottling, and
storing wines. Recipes given in British, U. S., and metric measures. Accompanying
booklet lists sources in U. S. A. where ingredients may be bought, and additional
information. 11 illustrations. 157pp. 5% x 8Vs-
(USO) 22090-7 Clothbound $3.50
Gardening With Herbs for Flavor and Fragrance, Helen M. Fox. How to
grow herbs in your own garden, how to use them in your cooking (over 55 recipes
included), legends and myths associated with each species, uses in medicine, per-
fumes, etc. — these are elements of one of the few books written especially for Amer-
ican herb fanciers. Guides you step-by-step from soil preparation to harvesting and
storage for each type of herb. 12 drawings by Louise Mansfield, xiv + 334pp.
22540-2 Paperbound $2.50
CATALOGUE OF DOVER BOOKS
American Food and Game Fishes, David S. Jordan and Barton W. Evermann.
Definitive source of information, detailed and accurate enough to enable the sports-
man and nature lover to identify conclusively some 1,000 species and sub-species
of North American fish, sought for food or sport. Coverage of range, physiology,
habits, life history, food value. Best methods of capture, interest to the angler, advice
on bait, fly-fishing, etc. 338 drawings and photographs. 1 + 574pp. 6% xV/fc.
22383-1 Paperbound $4.50
The Frog Book, Mary C. Dickerson. Complete with extensive finding keys, over
300 photographs, and an introduction to the general biology of frogs and toads, this
is the classic non-technical study of Northeastern and Central species. 58 species;
290 photographs and 16 color plates, xvii + 253pp.
21973-9 Paperbound $4.00
The Moth Book: A Guide to the Moths of North America, William J.
Holland. Classical study, eagerly sought after and used for the past 60 years. Clear
identification manual to more than 2,000 different moths, largest manual in existence.
General information about moths, capturing, mounting, classifying, etc., followed by
species by species descriptions. 263 illustrations plus 48 color plates show almost
every species, full size. 1968 edition, preface, nomenclature changes by A. E.
Brower. xxiv + 479pp. of text. 6y 2 x 9%.
21948-8 Paperbound $5.00
The Sea-Beach at Ebb-Tide, Augusta Foote Arnold. Interested amateur can iden-
tify hundreds of marine plants and animals on coasts of North America; marine
algae; seaweeds; squids; hermit crabs; horse shoe crabs; shrimps; corals; sea anem-
ones; etc. Species descriptions cover: structure; food; reproductive cycle; size;
shape; color; habitat; etc. Over 600 drawings. 85 plates, xii -f 490pp.
21949-6 Paperbound $3.50
Common Bird Songs, Donald J. Borror. 33 x /3 12-inch record presents songs of 60
important birds of the eastern United States. A thorough, serious record which pro-
vides several examples for each bird, showing different types of song, individual
variations, etc. Inestimable identification aid for birdwatcher. 32-page booklet gives
text about birds and songs, with illustration for each bird.
21829-5 Record, book, album. Monaural. $2.75
Fads and Fallacies in the Name of Science, Martin Gardner. Fair, witty ap-
praisal of cranks and quacks of science: Atlantis, Lemuria, hollow earth, flat earth,
Velikovsky, orgone energy, Dianetics, flying saucers, Bridey Murphy, food fads,
medical fads, perpetual motion, etc. Formerly "In the Name of Science." x + 363pp.
20394-8 Paperbound $2.00
Hoaxes, Curtis D. MacDougall. Exhaustive, unbelievably rich account of great
hoaxes: Locke's moon hoax, Shakespearean forgeries, sea serpents, Loch Nesj
monster, Cardiff giant, John Wilkes Booth's mummy, Disumbrationist school of art,
dozens more; also journalism, psychology of hoaxing. 54 illustrations, xi + 338pp.
20465-0 Paperbound $2.75
CATALOGUE OF DOVER BOOKS
Advanced Calculus, Edwin B. Wilson. Widely regarded as among the most
useful and comprehensive texts in this subject. Many chapters, such as those on
vector functions, ordinary differential equations, special functions, calculus of
variations, elliptic functions and partial differential equations, are excellent intro-
ductions to their branches of higher mathematics. More than 1300 exercises speed
comprehension and indicate applications, ix -f 566pp.
60504-3 Paperbound $3.00
A Treatise on Advanced Calculus, Philip Franklin. Comprehensive, logical
treatment of theory of calculus and allied subjects. Provides solid basis for gradu-
ate study without going as far as texts on real variable theory. Theory stressed over
applications and techniques. 612 exercise problems with solution hints. 28 figures.
xi -f 595pp. 61252-X Paperbound $4.00
Hydrodynamics, Sir Horace Lamb. Standard reference and study work, almost
inexhaustible in coverage of classical material. Unexcelled for fundamental theo-
rems, equations, detailed methods of solution: equations of motion, integration
of equations, irrotational motion, motion oi liquid in two dimensions, motion of
solids through liquids, vortex motion, tidal waves, waves of expansion, surface
waves, viscosity, rotating liquids, etc. 6th enlarged edition. 119 figures, xv +
738pp. 6x9. (USO) 60256-7 Paperbound $4.00
Electromagnetism, John C. Slater and Nathaniel H. Frank. Introductory study
by leading men in the field supplies basic material on electrostatics and magneto-
statics, then concentrates on electromagnetic theory, ranging over many areas and
touching on electrical engineering. Also covers equations and theorems of Gauss,
Poisson, Laplace and Green, dielectrics, magnetic fields of linear and circular
currents, electromagnetic induction and Maxwell's equations, wave guides and
cavity resonators, Huygens' principle, etc. A knowledge of calculus and differential
equations required. Problems are supplied. 39 figures, xii + 240pp.
62263-0 Paperbound $2.75
Applied Hydro- and Aeromechanics, Ludwig Prandtl, O. G. Tietjens. Methods
valuable to engineers: flow in pipes, boundary layers, airfoil theory, entry condi-
tions, turbulent flow in pipes, drag, etc. 226 figures, 287 photographic plates,
xvi + 311pp. 60375-X Paperbound $2.50
Basic Optics and Optical Instruments, U. S. Navy. Navy elementary train-
ing manual, clearly treating the composition of optical glass, characteristics of light,
elements of mirrors, prisms and lenses, construction of optical instruments, main-
tenance and repair procedures. Formerly titled Opticalman 2 & 3. Nearly 600
charts, diagrams, photgraphs and drawings, vi -f- 485pp. 6Y 2 x 9Y 4 .
22291-8 Paperbound $3.50
Mechanics of the Gyroscope: The Dynamics of Rotation, Richard F.
Diemel. Applications of gyroscopic phenomena stressed in this elementary treat-
ment of the dynamics of rotation. Covers velocity on a moving curve, gyroscopic
phenomena and apparatus, the gyro-compass, stabilizers (ships and monorail ve-
hicles). "Remarkably concise and generous treatment," Industrial Laboratories.
75 figures. 136 exercises, ix + 192pp. 60066-1 Paperbound $1.75
CATALOGUE OF DOVER BOOKS
Planets, Stars and Galaxies: Descriptive Astronomy for Beginners, A. E.
Fanning. Comprehensive introductory survey of astronomy: the sun, solar system,
stars, galaxies, universe, cosmology; up-to-date, including quasars, radio stars, etc.
Preface by Prof. Donald Menzel. 24pp. of photographs. 189pp. 5*4 x 8*4. '
21680-2 Paperbound $1.50
Teach Yourself Calculus, P. Abbott. With a good background in algebra and
trig, you can teach yourself calculus with this book. Simple, straightforward intro-
duction to functions of all kinds, integration, differentiation, series, etc. "Students
who are beginning to study calculus method will derive great help from this book."
Faraday House Journal. 308pp. 20683-1 Clothbound $2.00
Teach Yourself Trigonometry, P. Abbott. Geometrical foundations, indices and
logarithms, ratios, angles, circular measure, etc. are presented in this sound, easy-to-
use text. Excellent for the beginner or as a brush up, this text carries the student
through the solution of triangles. 204pp. 20682-3 Clothbound $2.00
Teach Yourself Anatomy, David LeVay. Accurate, inclusive, profusely illus-
trated account of structure, skeleton, abdomen, muscles, nervous system, glands,
brain, reproductive organs, evolution. "Quite the best and most readable account,'
Medical Officer. 12 color plates. 164 figures. 311pp. 4% x 7.
21651-9 Clothbound $2.50
Teach Yourself Physiology, David LeVay. Anatomical, biochemical bases ; di-
gestive, nervous, endocrine systems; metabolism; respiration; muscle; excretion;
temperature control; reproduction. "Good elementary exposition," The Lancet. 6
color plates. 44 illustrations. 208pp. 4y 4 x 7. 21658-6 Clothbound $2.50
The Friendly Stars, Martha Evans Martin. Classic has taught naked-eye observa-
tion of stars, planets to hundreds of thousands, still not surpassed for charm, lucidity,
adequacy. Completely updated by Professor Donald H. Menzel, Harvard Observa-
tory. 25 illustrations. 16 x 30 chart, x + 147pp. 21099-5 Paperbound $1.25
Music of the Spheres: The Material Universe from Atom to Quasar,
Simply Explained, Guy Murchie. Extremely broad, brilliantly written popular
account begins with the solar system and reaches to dividing line between matter and
nonmatter; latest understandings presented with exceptional clarity. Volume One:
Planets, stars, galaxies, cosmology, geology, celestial mechanics, latest astronomical
discoveries; Volume Two: Matter, atoms, waves, radiation, relativity, chemical
action, heat, nuclear energy, quantum theory, music, light, color, probability, anti-
matter, antigravity, and similar topics. 319 figures. 1967 (second) edition. Total
of xx + 644pp. 21809-0, 21810-4 Two volumes, Paperbound $5.00
Old-Time Schools and School Books, Clifton Johnson. Illustrations and rhymes
from early primers, abundant quotations from early textbooks, many anecdotes of
school life enliven this study of elementary schools from Puritans to middle 19th
century. Introduction by Carl Withers. 234 illustrations, xxxiii + 381pp.
21031-6 Paperbound $2.50
CATALOGUE OF DOVER BOOKS
Basic Electricity, U. S. Bureau of Naval Personel. Originally a training course,
best non-technical coverage of basic theory of electricity and its applications. Funda-
mental concepts, batteries, circuits, conductors and wiring techniques, AC and DC,
inductance and capacitance, generators, motors, transformers, magnetic amplifiers,
synchros, servomechanisms, etc. Also covers blue-prints, electrical diagrams, etc.
Many questions, with answers. 349 illustrations, x 4- 448pp. 6V^ x 9*4.
20973-3 Paperbound $3.00
Tensors For Circuits, Gabriel Kron. The purpose of this volume was to develop
a new mathematical method of analyzing engineering problems — through tensor
analysis — which has since proven its usefulness especially in electrical and structural
networks in computers. Introduction by Banesh Hoffmann. Formerly A Short
Course in Tensor Analysis. Over 800 figures, xviii -}- 250pp.
60534-5 Paperbound $2.00
Information Theory, Stanford Goldman. A thorugh presentation of the work
of C. E. Shannon and to a lesser extent Norbert Weiner, at a mathematical level
understandable to first-year graduate students in electrical engineering. In addi-
tion, the basic and general aspects of information theory are developed at an
elementary level for workers in non-mathematical sciences. Table of logarithms to
base 2. xiii -f 385pp. 62209-6 Paperbound $3.50
Introduction to the Statistical Dynamics of Automatic Control
Systems, V. V. Solodovnikov. General theory of control systems subjected to
random signals. Theory of linear analysis, statistics of random signals, theory of
linear prediction and filtering. For advanced and graduate-level students. Trans-
lated by John B. Thomas and Lotfi A. Zadeh. xxi + 307pp.
60420-9 Paperbound $3.00
Fundamental of Hydro- and Aeromechanics, Ludwig Prandtl and O. G.
Tietjens. Tietjens' famous expansion of Professor Prandtl's Kaiser Wilhelm
Institute lectures. Much original material included in coverage of statics of liquids
and gases, kinematics of liquids and gases, dynamics of non-viscous liquids. Proofs
are rigorous and use vector analysis. Translated by L. Rosenhead. 186 figures,
xvi + 270pp. 60374-1 Paperbound $2.25
Mathematical Methods for Scientists and Engineers, L. P. Smith. Full
investigation of methods, practical description of conditions where used: elements
of real functions, differential and integral calculus, space geometry, residues,
vectors and tensors, Bessel functions, etc. Many examples from scientific literature
completely worked out. 368 problems for solution, 100 diagrams, x + 453pp.
60220-6 Paperbound $2.75
Computational Methods of Linear Algebra, V. N. Faddeeva. Only work in
English to present classical and modern Russian computational methods of linear
algebra, including the work of A. N. Krylov, A. M. Danilevsky, D. K. Faddeev
and others. Detailed treatment of the derivation of numerical solutions to problems
of linear algebra. Translated by Curtis D. Benster. 23 carefully prepared tables.
New bibliography, x + 252pp. 60424-1 Paperbound $2.50
CATALOGUE OF DOVER BOOKS
Two Little Savages; Being the Adventures of Two Boys Who Lived as
Indians and What They Learned, Ernest Thompson Seton. Great classic of
nature and boyhood provides a vast range of woodlore in most palatable form, a
genuinely entertaining story. Two farm boys build a teepee in woods and live in it
for a month, working out Indian solutions to living problems, star lore, birds and
animals, plants, etc. 293 illustrations, vii + 286pp.
20985-7 Paperbound $2.50
Peter Piper's Practical Principles of Plain & Perfect Pronunciation.
Alliterative jingles and tongue-twisters of surprising charm, that made their first
appearance in America about 1830. Republished in full with the spirited woodcut
illustrations from this earliest American edition. 32pp. 4 l / 2 x 6 3 / 8 .
22560-7 Paperbound $1.00
Science Experiments and Amusements for Children, Charles Vivian. 73 easy
experiments, requiring only materials found at home or easily available, such as
candles, coins, steel wool, etc.; illustrate basic phenomena like vacuum, simple
chemical reaction, etc. All safe. Modern, well-planned. Formerly Science Games
for Children. 102 photos, numerous drawings. 96pp. 6% x 9%.
21856-2 Paperbound $1.25
An Introduction to Chess Moves and Tactics Simply Explained, Leonard
Barden. Informal intermediate introduction, quite strong in explaining reasons for
moves. Covers basic material, tactics, important openings, traps, positional play in
middle game, end game. Attempts to isolate patterns and recurrent configurations.
Formerly Chess. 58 figures. 102pp. (USO) 21210-6 Paperbound $1.25
Lasker's Manual of Chess, Dr. Emanuel Lasker. Lasker was not only one of the
five great World Champions, he was also one of the ablest expositors, theorists, and
analysts. In many ways, his Manual, permeated with his philosophy of battle, filled
with keen insights, is one of the greatest works ever written on chess. Filled with
analyzed games by the great players. A single-volume library that will profit almost
any chess player, beginner or master. 308 diagrams, xli x 349pp.
20640-8 Paperbound $2.75
The Master Book of Mathematical Recreations, Fred Schuh. In opinion of
many the finest work ever prepared on mathematical puzzles, stunts, recreations ;
exhaustively thorough explanations of mathematics involved, analysis of effects,
citation of puzzles and games. Mathematics involved is elementary. Translated by
F. Gobel. 194 figures, xxiv + 430pp. 22134-2 Paperbound $3.00
Mathematics, Magic and Mystery, Martin Gardner. Puzzle editor for Scientific
American explains mathematics behind various mystifying tricks: card tricks, stage
"mind reading," coin and match tricks, counting out games, geometric dissections,
etc. Probability sets, theory of numbers clearly explained. Also provides more than
400 tricks, guaranteed to work, that you can do. 135 illustrations, xii + 176pp.
20338-2 Paperbound $1.50
CATALOGUE OF DOVER BOOKS
Poems of Anne Bradstreet, edited with an introduction by Robert Hutchinson.
A new selection of poems by America's first poet and perhaps the first significant
woman poet in the English language. 48 poems display her development in works
of considerable variety — love poems, domestic poems, religious meditations, formal
elegies, "quaternions," etc. Notes, bibliography, viii + 222pp.
22160-1 Paperbound $2.00
Three Gothic Novels: The Castle of Otranto by Horace Walpole;
Vathek by William Beckford; The Vampyre by John Polidori, with Frag-
ment of a Novel by Lord Byron, edited by E. F. Bleiler. The first Gothic
novel, by Walpole; the finest Oriental tale in English, by Beckford; powerful
Romantic supernatural story in versions by Polidori and Byron. All extremely
important in history of literature; all still exciting, packed with supernatural
thrills, ghosts, haunted castles, magic, etc. xl + 291pp.
21232-7 Paperbound $2.00
The Best Tales of Hoffmann, E. T. A. HofTmann. 10 of Hoffmann's most
important stories, in modern re-editings of standard translations: Nutcracker and
the King of Mice, Signor Formica, Automata, The Sandman, Rath Krespel, The
Golden Flowerpot, Master Martin the Cooper, The Mines of Falun, The King's
Betrothed, A New Year's Eve Adventure. 7 illustrations by Hoffmann. Edited
by E. F. Bleiler. xxxix + 419pp. 21793-0 Paperbound $2.50
Ghost and Horror Stories of Ambrose Bierce, Ambrose Bierce. 23 strikingly
modern stories of the horrors latent in the human mind: The Eyes of the Panther,
The Damned Thing, An Occurrence at Owl Creek Bridge, An Inhabitant of Carcosa,
etc., plus the dream-essay, Visions of the Night. Edited by E. F. Bleiler. xxii
+ 1 99pp. 20767-6 Paperbound $1.50
Best Ghost Stories of J. S. LeFanu, J. Sheridan LeFanu. Finest stories by
Victorian master often considered greatest supernatural writer of all. Carmilia,
Green Tea, The Haunted Baronet, The Familiar, and 12 others. Most never before
available in the U. S. A. Edited by E. F. Bleiler. 8 illustrations from Victorian
publications, xvii + 467pp. 20415-4 Paperbound $2.50
The Time Stream, the Greatest Adventure, and the Purple Sapphire —
Three Science Fiction Novels, John Taine (Eric Temple Bell). Great Amer-
ican mathematician was also foremost science fiction novelist of the 1920's. The
Time Stream, one of all-time classics, uses concepts of circular time; The Greatest
Adventure, incredibly ancient biological experiments from Antarctica threaten to
escape; The Purple Sapphire, superscience, lost races in Central Tibet, survivors
of the Great Race. 4 illustrations by Frank R. Paul, v + 532pp.
21180-0 Paperbound $3.00
Seven Science Fiction Novels, H. G. Wells. The standard collection of the
great novels. Complete, unabridged. First Men in the Moon, Island of Dr. Moreau,
War of the Worlds, Food of the Gods, Invisible Man, Time Machine, In the Days
of the Comet. Not only science fiction fans, but every educated person owes it to
himself to read these novels. 1015pp. 20264-X Clothbound $5.00
CATALOGUE OF DOVER BOOKS
Tables of Functions: With Formulae and Curves, Eugene Jahnke and Fritz
Emde. The revised fourth edition, with the 76-page appendix of tables and
formulae of elementary functions. Sine, cosine, logarithmic integral; error integral;
Riemann-Zeta function; Mathieu functions; cubic equations; exponential func-
tion; the hyperbolic functions; much more. 212 figures, xii + 382pp.
(USO) 60133-1 Paperbound $2.50
Statistical Adjustment of Data, W. Edwards Deming. Introduction to basic
concepts of statistics, curve fitting, least squares solution, conditions without
parameter, conditions containing parameters. 26 exercises worked out. Some back-
ground in differential calculus desirable. Tables, x + 26lpp.
6123 5-X Paperbound $2.00
The Works of Archimedes, With the Method of Archimedes, edited by
T. L. Heath. All the known works of the great Greek mathematician. The editor's
186-page introduction describes the relation of Archimedes to his predecessors, his
life, and his thought. Supplement: The Method of Archimedes, recently discovered
by Heiberg. clxxxvi + 377pp. 60009-2 Paperbound $3.50
A Concise History of Mathematics, Dirk J. Struik. The best brief history of
mathematics. Stresses origins and covers every major figure in math history from
the ancient Near East to the great figures of the 19th century such as Fourier,
Gauss, Riemann, Cantor, many others. "A tremendous undertaking," American
Mathematical Monthly. Third revised edition. 41 illustrations, x + 195pp.
(EUK) 60255-9 Paperbound $2.00
An Introduction to Relaxation Methods, Frederick S. Shaw. Use of relaxa-
tion methods for solution of differential equations, written by co-worker with Sir
Richard Southwell, who developed relaxation methods. Deals with application of
a general computational process which has been extended to include almost all
branches of applied mechanics and physics. Treatment is mathematical rather than
physical. Detailed demonstrations and examples. 253 diagrams, 72 tables. 396pp.
60244-3 Paperbound $3-25
Mathematical Tables and Formulas, Robert D. Carmichael and Edwin R.
Smith. All tables necessary for college algebra and trigonometry. Five-place
logarithms, sines and tangents, trigonometric functions, powers, roots, reciprocals,
exponential and hyperbolic functions, formulas and theorems from geometry and
calculus. Very rich collection, viii + 269pp. 60111-0 Paperbound $1.50
Mathematical Tables of Elementary and Some Higher Mathematical
Functions, Herbert B. Dwight. Almost every function of importance in applied
mathematics, engineering and the physical sciences. Trigonometric functions and
their inverse functions to thousandths of radians, logarithms, hyperbolic functions,
elliptic functions of the first and second kind, over 60 pages of Bessel functions,
Euler numbers and their logs to base ten, and many others. Over half have
columns of difference to facilitate interpolation, viii -f- 231pp.
60445-4 Paperbound $2.50
CATALOGUE OF DOVER BOOKS
Planets, Stars and Galaxies: Descriptive Astronomy for Beginners, A. E.
Fanning. Comprehensive introductory survey of astronomy: the sun, solar system,
stars, galaxies, universe, cosmology; up-to-date, including quasars, radio stars, etc.
Preface by Prof. Donald Menzel. 24pp. of photographs. 189pp. 5% x 8Y 4 .
21680-2 Paperbound $1.50
Teach Yourself Calculus, P. Abbott. With a good background in algebra and
trig, you can teach yourself calculus with this book. Simple, straightforward intro-
duction to functions of all kinds, integration, differentiation, series, etc. "Students
who are beginning to study calculus method will derive great help from this book."
Faraday House Journal. 308pp. 20683-1 Clothbound $2.00
Teach Yourself Trigonometry, P. Abbott. Geometrical foundations, indices and
logarithms, ratios, angles, circular measure, etc. are presented in this sound, easy-to-
use text. Excellent for the beginner or as a brush up, this text carries the student
through the solution of triangles. 204pp. 20682-3 Clothbound $2.00
Teach Yourself Anatomy, David LeVay. Accurate, inclusive, profusely illus-
trated account of structure, skeleton, abdomen, muscles, nervous system, glands,
brain, reproductive organs, evolution. "Quite the best and most readable account,'
Medical Officer. 12 color plates. 164 figures. 311pp. 4y 4 x7.
21651-9 Clothbound $2.50
Teach Yourself Physiology, David LeVay. Anatomical, biochemical bases ; di-
gestive, nervous, endocrine systems; metabolism; respiration; muscle; excretion;
temperature control; reproduction. "Good elementary exposition," The Lancet. 6
color plates. 44 illustrations. 208pp. 4y 4 x 7. 21658-6 Clothbound $2.50
The Friendly Stars, Martha Evans Martin. Classic has taught naked-eye observa-
tion of stars, planets to hundreds of thousands, still not surpassed for charm, lucidity,
adequacy. Completely updated by Professor Donald H. Menzel, Harvard Observa-
tory. 25 illustrations. 16 x 30 chart. x+ 147pp. 2*1099-5 Paperbound $1.25
Music of the Spheres: The Material Universe from Atom to Quasar,
Simply Explained, Guy Murchie. Extremely broad, brilliantly written popular
account begins with the solar system and reaches to dividing line between matter and
nonmatter; latest understandings presented with exceptional clarity. Volume One:
Planets, stars, galaxies, cosmology, geology, celestial mechanics, latest astronomical
discoveries; Volume Two: Matter, atoms, waves, radiation, relativity, chemical
action, heat, nuclear energy, quantum theory, music, light, color, probability, anti-
matter, antigravity, and similar topics. 319 figures. 1967 (second) edition. Total
of xx + 644pp. 21809-0, 21810-4 Two volumes, Paperbound $5.00
Old-Time Schools and School Books, Clifton Johnson. Illustrations and rhymes
from early primers, abundant quotations from early textbooks, many anecdotes of
school life enliven this study of elementary schools from Puritans to middle 19th
century. Introduction by Carl Withers. 234 illustrations, xxxiii + 381pp.
21031-6 Paperbound $2.50
CATALOGUE OF DOVER BOOKS
Mechanics of Materials, Alvin Sloane. Over 500 problems are used to illustrate
the theory of elasticity introduced in this lucid text, ideal for class use. 299
figures. 553 problems, many answered. 17 tables, xvii + 468pp.
61767-X Paperbound $2.75
Answer Book: Detailed, fully worked solutions for all 553 problems. 9V 2 x 12y 4 .
26pp- 62513-3 Paperbound 75c
Applied Optics and Optical Design, A. E. Conrady. Standard work for
designers. Detailed step-by-step exposition, moving from fundamentals to design
of various systems. Volume 1 covers ordinary raytracing methods, primary aberra-
tion, telescopes, low-power microscope, photographic equipment. Volume 2 (com-
pleted by Rudolph Kingslake) covers high-power microscope, anastigmatic photo-
graphic objectives. 193 diagrams. Total of xiv -f 84lpp.
60611-2, 60612-2 Two volumes, Paperbound $7.50
Introduction to Electronics, U. S. Bureau of Naval Personnel. Basic con-
cepts, techniques, equipment on elementary level, no background required. Power
supplies, electron tubes in circuits, operation of transistors, servosystems, radio,
radar, sonar, etc. 155 figures, viii + 145pp. 6y 2 x 914.
21283-1 Paperbound $1.25
Basic Electronics, U. S. Bureau of Naval Personnel. Using nothing more ad-
vanced than elementary electricity and mathematics, this manual covers electron
tubes, circuits, antennas, radar, etc. — even transistors. 430 wiring diagrams, cut-
aways, photos, vii + 459pp. 6Y 2 x 9 l A- 21076-6 Paperbound $2.95
Basic Theory and Application of Transistors, U. S. Department of the
Army. Fundamental theory, applications for persons with minimal electronic
background. Physical basis, amplifiers, bias stabilization, analysis and comparison,
amplifiers, pulse and switching circuits, modulation, semiconductors, etc. Thorough,
nothing left out. 240 diagrams. 263pp. 6I/2 x 9V 4 .
20380-8 Paperbound $1.75
Microwave Electronics, John C. Slater. Full treatment of microwave elec-
tronics and its electromagnetic basis covers passive microwave circuits and active
devices such as klystrons, magnetrons, etc. A detailed mathematical analysis is
supplemented with experiments and physical pictures. Work is chiefly concerned
with basic physical principles rather than specific devices. 91 figures, xiv +
406pp. 62264-9 Paperbound $3.50
Handbook of Mathematical Functions With Formulas, Graphs, and
Mathematical Tables, edited by Milton Abramowitz and Irene A. Stegun.
Vast compendium of tables of functions designed to meet needs of scientists in all
fields. 29 sets of tables, some to as high as 20 places ; error function, Bessel func-
tions, Struve functions, Coulomb wave functions, hypergeometric functions, elliptic
integrals, Mathieu functions, orthogonal polynomials, probability functions, Laplace
transforms, etc. Originally published by U. S. Department of Commerce. Revised
edition, xiv + 1046pp. 8 x 10y 2 . 61272-4 Paperbound $5.00
CATALOGUE OF DOVER BOOKS
Johann Sebastian Bach, Philipp Spitta. One of the great classics of musicology,
this definitive analysis of Bach's music (and life) has never been surpassed. Lucid,
nontechnical analyses of hundreds of pieces (30 pages devoted to St. Matthew Pas-
sion, 26 to B Minor Mass). Also includes major analysis of 18th-century music.
450 musical examples. 40-page musical supplement. Total of xx + 1799pp.
(EUK) 22278-0, 22279-9 Two volumes, Clothbound $15.00
Mozart and His Piano Concertos, Cuthbert Girdlestone. The only full-length
study of an important area of Mozart's creativity. Provides detailed analyses of all
23 concertos, traces inspirational sources. 417 musical examples. Second edition.
509pp. (USO) 21271-8 Paperbound $3.50
The Perfect Wagnerite: A Commentary on the Niblung's Ring, George
Bernard Shaw. Brilliant and still relevant criticism in remarkable essays on
Wagner's Ring cycle, Shaw's ideas on political and social ideology behind the
plots, role of Leitmotifs, vocal requisites, etc. Prefaces, xxi -f 136pp.
21707-8 Paperbound $1.50
Don Giovanni, W. A. Mozart. Complete libretto, modern English translation;
biographies of composer and librettist; accounts of early performances and critical
reaction. Lavishly illustrated. All the material you need to understand and
appreciate this great work. Dover Opera Guide and Libretto Series; translated
and introduced by Ellen Bleiler. 92 illustrations. 209pp.
21134-7 Paperbound $1.50
High Fidelity Systems: A Layman's Guide, Roy F. Allison. All the basic
information you need for setting up your own audio system: high fidelity and
stereo record players, tape records, F.M. Connections, adjusting tone arm, cartridge,
checking needle alignment, positioning speakers, phasing speakers, adjusting hums,
trouble-shooting, maintenance, and similar topics. Enlarged 1965 edition. More
than 50 charts, diagrams, photos, iv -f 91pp. 21514-8 Paperbound $1.25
Reproduction of Sound, Edgar Villchur. Thorough coverage for laymen of
high fidelity systems, reproducing systems in general, needles, amplifiers, preamps,
loudspeakers, feedback,, explaining physical background. "A rare talent for making
technicalities vividly comprehensible," R. Darrell, High Fidelity. 69 figures.
iv-f92pp. 21515-6 Paperbound $1.00
Hear Me Talkin' to Ya: The Story of Jazz as Told by the Men Who
Made It, Nat Shapiro and Nat Hentoff. Louis Armstrong, Fats Waller, Jo Jones,
Clarence Williams, Billy Holiday, Duke Ellington, Jelly Roll Morton and dozens
of other jazz greats tell how it was in Chicago's South Side, New Orleans, depres-
sion Harlem and the modern West Coast as jazz was born and grew, xvi -j- 429pp.
21726-4 Paperbound $2.50
Fables of Aesop, translated by Sir Roger L'Estrange. A reproduction of the very
rare 1931 Paris edition; a selection of the most interesting fables, together with 50
imaginative drawings by Alexander Calder. v + 128pp. 6 1 / 2 x9 1 / 4-
21780-9 Paperbound $1.25
CATALOGUE OF DOVER BOOKS
Almost Periodic Functions, A. S. Besicovitch. Thorough summary of Bohr's
theory of almost periodic functions citing new shorter proofs, extending the theory,
and describing contributions of Wiener, Weyl, de la Vallee, Poussin, Stepanoff,
Bochner and the author, xiii + 180pp. 60018-1 Paperbound $1.75
An Introduction to The Study of Stellar Structure, S. Chandrasekhar.
A rigorous examination, using both classical and modern mathematical methods, of
the relationship between loss of energy, the mass, and the radius of stars in a steady
state. 38 figures. 509pp. 60413-6 Paperbound $3.25
Introduction to the Theory of Group's of Finite Order, Robert D. Car-
michael. Progresses in easy steps from sets, groups, permutations, isomorphism
through the important types of groups. No higher mathematics is necessary. 783
exercises and problems, xiv + 447pp. 60300-8 Paperbound $3.50
The Solubility of Nonelectrolytes, Joel H. Hildebrand and Robert L. Scott.
Classic, pioneering work discusses in detail ideal and nonideal solutions, inter-
molecular forces, structure of liquids, athermal mixing, hydrogen bonding, equa-
tions describing mixtures of gases, high polymer solutions, surface phenomena, etc.
Originally published in the American Chemical Society Monograph series. New
authors' preface and new paper (1964). 148 figures, 88 tables, xiv + 488pp.
61125-6 Paperbound $3.00
Introduction to Applied Mathematics, Francis D. Murnaghan. Introduction
to advanced mathematical techniques — vector and matrix analysis, partial differen-
tial equations, integral equations, Laplace transform theory, Fourier series,
boundary- value problems, etc. — particularly useful to physicists and engineers. 4l
figures, ix + 389pp. 61042-X Paperbound $2.25
Elementary Mathematics From An Advanced Standpoint: Volume I —
Arithmetic, Algebra, Analysis, Felix Klein. Second-level approach, illumi-
nated by graphical and geometrical interpretation. Covers natural and complex
numbers, real equations with real unknowns, equations in the field of complex
quantities, logarithmic and exponential functions, goniometric functions, infini-
tesimal calculus, transcendence of e and ir. Concept of function introduced im-
mediately. Translated by E. R. Hedrick and C. A. Noble. 125 figures, ix + 274pp.
(USO) 60150-1 Paperbound $2.25
Elementary Mathematics From An Advanced Standpoint: Volume II —
Geometry, Feliex Klein. Using analytical formulas, Klein clarifies the precise
formulation of geometric facts in chapters on manifolds, geometric and higher
point transformations, foundations. "Nothing comparable," Mathematics Teacher.
Translated by E. R. Hedrick and C. A. Noble. 141 figures, ix -f 214pp.
(USO) 60151-X Paperbound $2.25
Engineering Mathematics, Kenneth S. Miller. Most useful mathematical tech-
niques for graduate students in engineering, physics, covering linear differential
equations, series, random functions, integrals, Fourier series, Laplace transform,
network theory, etc. "Sound and teachable," Science. 89 figures, xii + 4l7pp.
6x8l/ 2 . 61121-3 Paperbound $3.00
CATALOGUE OF DOVER BOOKS
Introduction to Astrophysics: The Stars, Jean Dufay. Best guide to ob-
servational astrophysics in English. Bridges the gap between elementary populariza-
tions and advanced technical monographs. Covers stellar photometry, stellar spectra
and classification, Hertzsprung-Russell diagrams, Yerkes 2-dimensional classifica-
tion, temperatures, diameters, masses and densities, evolution of the stars. Trans-
lated by Owen Gingerich. 51 figures, 11 tables, xii -f 164pp.
(USCO) 60771-2 Paperbound $2.00
Introduction to Bessel Functions, Frank Bowman. Full, clear introduction to
properties and applications of Bessel functions. Covers Bessel functions of zero
order, of any order; definite integrals ; asymptotic expansions ; Bessel' s solution to
Kepler's problem; circular membranes ; etc. Math above calculus and fundamentals
of differential equations developed within text. 636 problems. 28 figures, x +
135pp. 60462-4 Paperbound $1.75
Differential and Integral Calculus, Philip Franklin. A full and basic intro-
duction, textbook for a two- or three-semester course, or self-study. Covers para-
metric functions, force components in polar coordinates, Duhamel's theorem,
methods and applications of integration, infinite series, Taylor's series, vectors and
surfaces in space, etc. Exercises follow each chapter with full solutions at back
of the book. Index, xi + 679pp. 62520-6 Paperbound $4.00
The Exact Sciences in Antiquity, O. Neugebauer. Modern overview chiefly
of mathematics and astronomy as developed by the Egyptians and Babylonians.
Reveals startling advancement of Babylonian mathematics (tables for numerical
computations, quadratic equations with two unknowns, implications that Pytha-
gorean theorem was known 1000 years before Pythagoras), and sophisticated
astronomy based on competent mathematics. Also covers transmission of this
knowledge to Hellenistic world. 14 plates, 52 figures, xvii + 240pp.
22332-9 Paperbound $2.50
The Thirteen Books of Euclid's Elements, translated with introduction and
commentary by Sir Thomas Heath. Unabridged republication of definitive edition
based on the text of Heiberg. Translator's notes discuss textual and linguistic
matters, mathematical analysis, 2500 years of critical commentary on the Elements.
Do not confuse with abridged school editions. Total of xvii + 1414pp.
60088-2, 60089-0, 60090-4 Three volumes, Paperbound $8.50
An Introduction to Symbolic Logic, Susanne K. Langer. Well-known intro-
duction, popular among readers with elementary mathematical background. Starts
with simple symbols and conventions and teaches Boole-Schroeder and Russell-
Whitehead systems. 367pp. 60164-1 Paperbound $2.25
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Guide to the Literature of Mathematics and Physics,
Nathan G. Parke III. (60447-0) $3.00
A Survey or Physical Theory, Max Planck. (60650-3) $1.50
The Theory of Heat Radiation, Max Planck. (60546-9) $2.50
Treatise on Thermodynamics, Max Planck, (60219-2) $2.50
Scientific Papers, Lord Rayleigh (John William Strutt).
(61213-9, 61214-7. 61215-5) Three-volume set, Cloth-
bound $37.50
The Theory of Sound, Lord Rayleigh (John William Strutt).
(60292-3, 60293-1) Two-volume set $6.00
Radiochemistry and the Discovery of Isotopes, edited by
Alfred Romer. (62507-9) $3.50
The Discovery of Radioactivity and Transmutation, edited
by Alfred Romer. (61198-1) $2.50
Selected Papers on Cosmic Ray Origin Theories, edited by
Stephen Rosen. (62127-8) $5.00
History of Hydraulics, Hunter Rouse and Simon Ince.
(61131-0) $2.50
The Analysis of Matter, Bertrand Russell. (60231-1) $3.00
Essay on the Foundations of Geometry, Bertrand Russell.
(60233-8) $2.50
The Study of the History oi Mathematics and The Study
of the History of Science, George Sarton. (20240-2)
$2.25
History of Mathematics, David Eugene Smith. (20429-4,
20430-8) Two-volume set $8.00
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THE HISTORY OF
THE CALCULUS AND ITS
CONCEPTUAL DEVELOPMENT
CARL B. BOYER
This book., for the first time., provides laymen and mathematicians alike with a detailed
picture of the historical development of one of the most momentous achievements of the
human intellect— the calculus. It describes with accuracy and perspective the long
development of both the integral and the differential calculus from their early beginnings
in antiquity to their final emancipation in the 19th century from both physical and
metaphysical ideas alike and their final elaboration as mathematical abstractions, as we
know them today, defined in terms of formal logic by means of the idea of a limit of
an infinite sequence. ^
But while the importance of the calculus and mathematical analysis — the core of modern
mathematics cannot be overemphasized, the value of this first comprehensive critical
history of the calculus goes far beyond the subject matter. This book will fully counteract
the impression of laymen, and of many mathematicians, that the great achievements of
mathematics were formulated from the beginning in final form. It will give readers a
sense of mathematics not as a technique, but as a habit of mind, and serve to bridge the
gap between the sciences and the humanities, It will also make abundantly clear the
modern understanding of mathematics by showing in detail how the concepts of the
calculus gradually changed from the Greek view of the reality and immanence of
mathematics to the revised concept of mathematical rigor developed by the great 19th
century mathematicians, which held that any premises were valid so long as they were
consistent with the one another, It will make clear the ideas contributed by Zeno, Plato,
Pythagoras, Eudoxus, the Arabic and Scholastic mathematicians, Newton, Leibnitz, Taylor,
Descartes, Euier, Lagrange, Cantor, Weierstrass and many others in the long passage
from the Greek "method of exhaustion" and Zeno's paradoxes to the modern concept of
the limit independent of sense experience; and illuminate not only the methods of
mathematical discovery, but the foundations of mathematical thought as we
Complete, unabridged republication. Foreword by R. Courant. Preface. 22 figures. 25-page
bibliography. Index, v + 346pp. %% x 8. 60509-4 Paperbound
A DOVER EDITION DESIGNED FDR YEARS OF USE!
We have made every effort to make this the best hook possible. Our paper is opaque, with
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happens with paperbacks held together with glue. Books open flat for easy reference. The
binding will not crack or split. This is a permanent book.
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